diff --git a/LANGUAGE.md b/LANGUAGE.md
index 2e573fe..968ab1c 100644
--- a/LANGUAGE.md
+++ b/LANGUAGE.md
@@ -39,8 +39,8 @@ roadmap and milestone history.
 
 ### expressions and control flow
 
-- checked integer `+ - * /`, unary `-`, divide-by-zero traps, IEEE float arithmetic, comparisons, `!`, `and`, and `or`
-- assignments and compound assignments `+= -= *= /=` with single evaluation of complex lvalues
+- checked integer `+ - *`, unary `-`, float-only `/`, IEEE float arithmetic, comparisons, `!`, `and`, and `or`
+- assignments and compound assignments `+= -= *= /=` with single evaluation of complex lvalues; `/=` is float-only
 - field access through struct values and pointers, index/slice bounds contextually coerced to `usize`, and unsigned narrower index support
 - boolean `if` / `else if` / `else`, braceless single-statement branches, and optional parenthesized conditions
 - `while` loops with optional post-iteration update clauses
@@ -52,6 +52,50 @@ roadmap and milestone history.
 - fallible `try`, fallback `catch`, and `catch |e| { ... }` handler blocks
 - direct `return match ...` and `yield match ...` value-control-flow operands
 
+#### division
+
+`/` and `/=` accept only floating-point operands. Integer division must state its rounding and
+remainder convention with one of these unqualified builtins:
+
+| Builtin | Result |
+| --- | --- |
+| `div_trunc(a, b)` | quotient rounded toward zero |
+| `div_floor(a, b)` | quotient rounded toward negative infinity |
+| `div_exact(a, b)` | truncated quotient; traps unless it divides exactly |
+| `div_ceil(a, b)` | quotient rounded toward positive infinity |
+| `rem(a, b)` | remainder paired with `div_trunc`; sign follows `a` |
+| `mod(a, b)` | modulus paired with `div_floor`; sign follows `b` |
+
+The operands may be compatible concrete integer or float scalars. Existing literal coercion and
+numeric widening rules apply, the result has the common operand type, and float quotients are
+integral-valued floats. These identities hold when representable:
+
+```bro
+div_trunc(a, b) * b + rem(a, b) == a
+div_floor(a, b) * b + mod(a, b) == a
+```
+
+Negative operands distinguish the operations:
+
+```bro
+div_trunc(-5, 3) == -1
+div_floor(-5, 3) == -2
+div_ceil(-5, 3) == -1
+rem(-5, 3) == -2
+mod(-5, 3) == 1
+mod(5, -3) == -1
+```
+
+All six builtins diagnose a zero denominator at comptime and trap at runtime, including float
+zero. Quotient operations also trap for signed `min_value(T), -1`; `rem` and `mod` return zero for
+that pair. `div_exact` traps when `div_trunc(a, b) * b == a` is false in the operand type, so float
+exactness follows floating-point equality. Other float NaN and infinity behavior follows the
+underlying IEEE operations. Ordinary float `/` remains unchecked and therefore preserves IEEE
+infinity/NaN behavior.
+
+The six spellings are reserved only as direct unqualified calls. A qualified call such as
+`math.div_floor(a, b)` resolves to an ordinary package function.
+
 ### functions, C interop, and linking
 
 - demand-monomorphized Brolang and C-ABI functions
diff --git a/README.md b/README.md
index 340a95e..793b7df 100644
--- a/README.md
+++ b/README.md
@@ -187,6 +187,7 @@ Current prototype features:
 - Bodyless manual and imported C variadic declarations with default argument promotions
 - Ordered linking of additional C sources, objects, archives, and libraries
 - Checked signed addition and unary negation
+- Float-only `/` plus explicit `div_trunc`, `div_floor`, `div_exact`, `div_ceil`, `rem`, and `mod` scalar builtins
 - Static, eager runtime, mutable runtime, and deferred problematic globals
 - Runtime diagnostics followed by `llvm.trap`
 
diff --git a/TODO.md b/TODO.md
index bb47574..be8a802 100644
--- a/TODO.md
+++ b/TODO.md
@@ -116,7 +116,7 @@
     - `for 0..(len) |i| { ... }` or equivalently `for 0..=(len - 1) |i| { ... }` - calculating range bounds, expressions must be parenthesized
   - for all conditionals/guards, parentheses are optional but allowed for visual clarity
 
-6. compound assignment: `+=`, `-=`, `*=`, `/=` (implemented)
+6. compound assignment: `+=`, `-=`, `*=`, `/=` (implemented; division semantics superseded by milestone 32)
   - added the binary arithmetic operators `-`, `*`, `/` (previously only `+` existed); `*`/`/`
     bind tighter than `+`/`-`, and prefix `-` (negation) is unchanged
   - compound assignments preserve the target, operator, and right-hand side explicitly through
@@ -124,10 +124,9 @@
     operation, and stores through that address
   - side-effecting index, field-base, and dereference expressions are evaluated once in
     left-to-right order
-  - integer arithmetic traps on overflow (`Sub_Checked`/`Mul_Checked` via the LLVM
-    `.with.overflow` intrinsics) and integer `/` traps on divide-by-zero and `INT_MIN / -1`;
-    floats follow IEEE (`fadd`/`fsub`/`fmul`/`fdiv`, no trap)
-  - constant folding (global initializers) covers `-`, `*`, `/` alongside `+`
+  - integer `+`, `-`, and `*` trap on overflow; milestone 32 later restricted `/` and `/=` to
+    floats and introduced the explicit integer/float division family
+  - constant folding (global initializers) covers the arithmetic family
 
 7. enums (native and c interop) (implemented; see below)
   - native enums are nominal value types with integer runtime representations
@@ -791,10 +790,24 @@
   - the existing specialization/HIR/LLVM ABI is unchanged; `std/mem` and `std/arraylist` now use the
     inferred form where their arguments or result provide enough information
 
-32. disallow arbitrary integer division
-  - take inspiration from zig
-  - see also below for a word on unchecked casts
-  - the user should be explicit about what they mean with integer division (e.g. `div`, `rem`)
+32. explicit division family (implemented)
+  - `/` and `/=` are float-only; every integer use is rejected with guidance toward explicit
+    division, including literals, comptime execution, array counts, and compound assignment
+  - direct unqualified calls reserve `div_trunc`, `div_floor`, `div_exact`, `div_ceil`, `rem`, and
+    `mod`; qualified names remain ordinary package functions
+  - the builtins accept compatible concrete integer or float scalars, reuse existing literal and
+    widening rules, and return the common operand type (integral-valued floats for quotients)
+  - all builtins diagnose zero denominators at comptime and trap at runtime; quotient operations
+    also trap on signed `min_value(T) / -1`, while `rem` and `mod` return zero for that pair
+  - `div_exact` checks the reconstructed dividend in the operand type; `rem` pairs with truncation
+    and follows the numerator sign, while `mod` pairs with floor and follows the denominator sign
+  - HIR/IR use compact semantic enum tags; integer floor, ceil, and exact lowering reconstructs the
+    remainder from one quotient so each produces only one hardware-division candidate
+  - float lowering uses the typed LLVM trunc/floor/ceil intrinsics, `frem`, and ordered equality;
+    ordinary float `/` remains the unchecked IEEE infinity/NaN escape hatch
+  - migrated `std/mem`, `std/arraylist`, and the compound-assignment example to `div_trunc`
+
+33. design io interface
 
 ## A word on unchecked casts
 
diff --git a/compiler/checker/checker.odin b/compiler/checker/checker.odin
index 8e13a20..db53a71 100644
--- a/compiler/checker/checker.odin
+++ b/compiler/checker/checker.odin
@@ -399,6 +399,31 @@ Type_Builtin :: enum u8 {
 	Max_Value,
 }
 
+Division_Builtin :: enum u8 {
+	None,
+	Trunc,
+	Floor,
+	Exact,
+	Ceil,
+	Rem,
+	Mod,
+}
+
+division_builtin_call :: proc(checker: ^Checker, expr: ast.Expr) -> Division_Builtin {
+	if expr.kind != .Call || expr.left != ast.INVALID_EXPR || symbol.is_valid(expr.qualifier) {
+		return .None
+	}
+	switch symbol_text(checker, expr.name) {
+	case "div_trunc": return .Trunc
+	case "div_floor": return .Floor
+	case "div_exact": return .Exact
+	case "div_ceil":  return .Ceil
+	case "rem":       return .Rem
+	case "mod":       return .Mod
+	}
+	return .None
+}
+
 type_builtin_call :: proc(checker: ^Checker, expr: ast.Expr) -> Type_Builtin {
 	if expr.kind != .Call || expr.left != ast.INVALID_EXPR || symbol.is_valid(expr.qualifier) {
 		return .None
@@ -598,6 +623,10 @@ type_from_syntax :: proc(
 					changed = true
 				}
 			} else {
+				if constant.kind == .Integer_Division {
+					source.add(checker.diagnostics, span, "integer '/' is not allowed; use div_trunc, div_floor, div_exact, or div_ceil")
+					return types.INVALID
+				}
 				source.add(checker.diagnostics, span, "array count must be a compile-time integer expression")
 				return types.INVALID
 			}
@@ -2568,6 +2597,35 @@ infer_nested_expr :: proc(
 	return result
 }
 
+infer_division_builtin :: proc(
+	checker: ^Checker,
+	expr: ast.Expr,
+	locals: []Infer_Local,
+	pkg: ast.Package_Id,
+	file: ast.File_Id,
+	demanded: ^[dynamic]Spec_Id,
+	local_types: []types.Type,
+	expected: types.Type,
+) -> types.Type {
+	if len(expr.args) != 2 {
+		return types.INVALID
+	}
+	hint := expected if types.is_concrete_scalar(expected) && !types.is_bool(expected) else types.INVALID
+	left_const := is_numeric_constant_expr(checker, expr.args[0])
+	right_const := is_numeric_constant_expr(checker, expr.args[1])
+	left, right := types.INVALID, types.INVALID
+	if left_const && !right_const && !types.is_valid(hint) {
+		right = infer_nested_expr(checker, expr.args[1], locals, pkg, file, demanded, local_types)
+		left = infer_nested_expr(checker, expr.args[0], locals, pkg, file, demanded, local_types, right)
+	} else {
+		left = infer_nested_expr(checker, expr.args[0], locals, pkg, file, demanded, local_types, hint)
+		right_hint := hint if types.is_valid(hint) else left
+		right = infer_nested_expr(checker, expr.args[1], locals, pkg, file, demanded, local_types, right_hint)
+	}
+	result := types.widest(left, right)
+	return result if types.is_concrete_scalar(result) && !types.is_bool(result) else types.INVALID
+}
+
 infer_compound_expr :: proc(
 	checker: ^Checker,
 	expr: ast.Expr,
@@ -2938,6 +2996,11 @@ infer_expr :: proc(
 					_ = pop(&stack)
 					continue
 				}
+				if division_builtin_call(checker, expr) != .None {
+					last = infer_division_builtin(checker, expr, locals, pkg, file, demanded, local_types, frame.expected)
+					_ = pop(&stack)
+					continue
+				}
 				if is_ptr_cast_call(checker, expr) {
 					if len(expr.args) != 2 {
 						last = types.INVALID
@@ -3952,6 +4015,12 @@ record_demand :: proc(
 			right := record_demand(checker, expr.right, demand, locals, local_types, pkg, file)
 			return left || right
 		}
+	case .Call:
+		if division_builtin_call(checker, expr) != .None && len(expr.args) == 2 && is_numeric_demand(demand, checker.target) {
+			left := record_demand(checker, expr.args[0], demand, locals, local_types, pkg, file)
+			right := record_demand(checker, expr.args[1], demand, locals, local_types, pkg, file)
+			return left || right
+		}
 	}
 	return false
 }
@@ -4435,6 +4504,14 @@ build_constant_expr :: proc(
 		id := source.add(checker.diagnostics, expr.span, "division by zero in constant expression")
 		return invalid_hir_expr(checker, expr.span, id, recovery_type)
 	}
+	if constant.kind == .Non_Exact {
+		id := source.add(checker.diagnostics, expr.span, "exact division has a remainder")
+		return invalid_hir_expr(checker, expr.span, id, recovery_type)
+	}
+	if constant.kind == .Integer_Division {
+		id := source.add(checker.diagnostics, expr.span, "integer '/' is not allowed; use div_trunc, div_floor, div_exact, or div_ceil")
+		return invalid_hir_expr(checker, expr.span, id, recovery_type)
+	}
 	if constant.kind == .Overflow ||
 	   (!types.is_concrete_integer(expected) && !fits_i64(constant.value)) {
 		id := source.add(
@@ -4868,6 +4945,89 @@ build_nested_expr :: proc(
 	return result
 }
 
+try_build_comptime_division :: proc(
+	checker: ^Checker,
+	expr: ast.Expr,
+	kind: Division_Builtin,
+	expected: types.Type,
+	pkg: ast.Package_Id,
+	file: ast.File_Id,
+) -> (hir.Expr_Id, bool) {
+	state := ct_state_make(checker, pkg, file, diagnose=false)
+	defer ct_state_destroy(&state)
+	value, flow, ok := ct_eval_division_call(&state, expr, kind, expected, 0)
+	if ok && flow.kind == .Normal && value != INVALID_CT_VALUE {
+		return ct_materialize_value(&state, value, expr.span, expected), true
+	}
+	message := ""
+	#partial switch state.error {
+	case .Div_By_Zero: message = "division builtin denominator is zero"
+	case .Overflow:    message = "signed integer division overflow"
+	case .Non_Exact:   message = "exact division has a remainder"
+	}
+	if len(message) == 0 {
+		return hir.INVALID_EXPR, false
+	}
+	id := source.add(checker.diagnostics, expr.span, message)
+	recovery := expected if types.is_concrete_scalar(expected) else types.I64
+	return invalid_hir_expr(checker, expr.span, id, recovery), true
+}
+
+build_division_builtin :: proc(
+	checker: ^Checker,
+	expr: ast.Expr,
+	kind: Division_Builtin,
+	locals: []Build_Local,
+	global_reads: ^[dynamic]hir.Global_Id,
+	calls: ^[dynamic]hir.Function_Id,
+	expected: types.Type,
+	pkg: ast.Package_Id,
+	file: ast.File_Id,
+) -> hir.Expr_Id {
+	if len(expr.args) != 2 {
+		id := source.addf(
+			checker.diagnostics, expr.span, "%s expects 2 arguments, got %d",
+			symbol_text(checker, expr.name), len(expr.args),
+		)
+		return invalid_hir_expr(checker, expr.span, id)
+	}
+	if value, handled := try_build_comptime_division(checker, expr, kind, expected, pkg, file); handled {
+		return value
+	}
+	hint := expected if types.is_concrete_scalar(expected) && !types.is_bool(expected) else types.INVALID
+	left_const := is_numeric_constant_expr(checker, expr.args[0])
+	right_const := is_numeric_constant_expr(checker, expr.args[1])
+	left, right := hir.INVALID_EXPR, hir.INVALID_EXPR
+	if left_const && !right_const && !types.is_valid(hint) {
+		right = build_nested_expr(checker, expr.args[1], locals, global_reads, calls, types.INVALID, pkg, file)
+		left = build_nested_expr(checker, expr.args[0], locals, global_reads, calls, checker.module.exprs[right].type, pkg, file)
+	} else {
+		left = build_nested_expr(checker, expr.args[0], locals, global_reads, calls, hint, pkg, file)
+		right_hint := hint if types.is_valid(hint) else checker.module.exprs[left].type
+		right = build_nested_expr(checker, expr.args[1], locals, global_reads, calls, right_hint, pkg, file)
+	}
+	result := types.widest(checker.module.exprs[left].type, checker.module.exprs[right].type)
+	if !types.is_concrete_scalar(result) || types.is_bool(result) {
+		id := source.add(checker.diagnostics, expr.span, "division builtins require compatible numeric operands")
+		return invalid_hir_expr(checker, expr.span, id)
+	}
+	left = coerce_expr(checker, left, result, checker.module.exprs[left].span)
+	right = coerce_expr(checker, right, result, checker.module.exprs[right].span)
+	result_kind := hir.Expr_Kind.Div_Trunc
+	#partial switch kind {
+	case .Floor: result_kind = .Div_Floor
+	case .Exact: result_kind = .Div_Exact
+	case .Ceil:  result_kind = .Div_Ceil
+	case .Rem:   result_kind = .Rem
+	case .Mod:   result_kind = .Mod
+	case:
+	}
+	return add_hir_expr(checker, hir.Expr{
+		kind=result_kind, span=expr.span, type=result, left=left, right=right,
+		target=hir.INVALID_REF, diagnostic=source.INVALID_DIAGNOSTIC,
+	})
+}
+
 fallible_aggregate :: proc(
 	checker: ^Checker,
 	span: source.Span,
@@ -5578,6 +5738,13 @@ build_binary_arith :: proc(
 		id := source.add(checker.diagnostics, span, "arithmetic requires compatible numeric operands")
 		return invalid_hir_expr(checker, span, id)
 	}
+	if op == .Div && !types.is_float(result, checker.target) {
+		id := source.add(
+			checker.diagnostics, span,
+			"integer '/' is not allowed; use div_trunc, div_floor, div_exact, or div_ceil",
+		)
+		return invalid_hir_expr(checker, span, id, result)
+	}
 	result_kind := hir.Expr_Kind.Add
 	#partial switch op {
 	case .Sub: result_kind = .Sub
@@ -5627,7 +5794,7 @@ build_expr :: proc(
 		expr := checker.ast_module.exprs[frame.expr]
 		if frame.stage == 0 {
 			constant := eval_constant(checker, frame.expr)
-			if constant.kind == .Value || constant.kind == .Overflow || constant.kind == .Div_By_Zero {
+			if constant.kind == .Value || constant.kind == .Overflow || constant.kind == .Div_By_Zero || constant.kind == .Non_Exact {
 				last = build_constant_expr(checker, expr, constant, frame.expected)
 				_ = pop(&stack)
 				continue
@@ -5807,6 +5974,13 @@ build_expr :: proc(
 					_ = pop(&stack)
 					continue
 				}
+				if builtin := division_builtin_call(checker, expr); builtin != .None {
+					last = build_division_builtin(
+						checker, expr, builtin, locals, global_reads, calls, frame.expected, pkg, file,
+					)
+					_ = pop(&stack)
+					continue
+				}
 				if is_ptr_cast_call(checker, expr) {
 					if len(expr.args) != 2 {
 						id := source.addf(checker.diagnostics, expr.span, "ptr_cast expects 2 arguments, got %d", len(expr.args))
@@ -6653,7 +6827,14 @@ build_block :: proc(
 						}
 						rhs_type := checker.module.exprs[value].type
 						result_type := types.widest(target_type, rhs_type)
-						if !types.is_concrete_scalar(result_type) ||
+						if statement.assignment_op == .Div && types.is_concrete_integer(result_type) {
+							id := source.add(
+								checker.diagnostics,
+								statement.span,
+								"integer '/=' is not allowed; assign through an explicit division builtin",
+							)
+							value = invalid_hir_expr(checker, statement.span, id, target_type)
+						} else if !types.is_concrete_scalar(result_type) ||
 						   types.is_bool(result_type) {
 							id := source.add(
 								checker.diagnostics,
diff --git a/compiler/checker/comptime.odin b/compiler/checker/comptime.odin
index 3b29fce..8af6735 100644
--- a/compiler/checker/comptime.odin
+++ b/compiler/checker/comptime.odin
@@ -6,6 +6,8 @@ import "../source"
 import "../symbol"
 import "../types"
 import "base:intrinsics"
+
+import "core:math"
 import "core:mem"
 
 COMPTIME_EVAL_QUOTA :: 100_000
@@ -28,6 +30,8 @@ Constant_Kind :: enum {
 	Value,
 	Overflow,
 	Div_By_Zero,
+	Non_Exact,
+	Integer_Division,
 }
 
 Constant :: struct {
@@ -94,8 +98,7 @@ eval_constant :: proc(checker: ^Checker, expr_id: ast.Expr_Id) -> Constant {
 			continue
 		}
 		expr := checker.ast_module.exprs[frame.expr]
-		if expr.kind != .Add && expr.kind != .Sub && expr.kind != .Mul &&
-		   expr.kind != .Div && expr.kind != .Negate {
+		if expr.kind != .Add && expr.kind != .Sub && expr.kind != .Mul && expr.kind != .Negate {
 			result := Constant{kind = .Not_Constant}
 			if expr.kind == .Integer {
 				result = Constant{kind = .Value, value = i128(expr.integer)}
@@ -118,9 +121,7 @@ eval_constant :: proc(checker: ^Checker, expr_id: ast.Expr_Id) -> Constant {
 				operand = checker.constants[expr.left]
 			}
 			result := Constant{kind = .Not_Constant}
-			if operand.kind == .Div_By_Zero {
-				result = Constant{kind = .Div_By_Zero}
-			} else if operand.kind == .Overflow {
+			if operand.kind == .Overflow {
 				result = Constant{kind = .Overflow}
 			} else if operand.kind == .Value {
 				value, overflow := intrinsics.overflow_sub(i128(0), operand.value)
@@ -147,29 +148,19 @@ eval_constant :: proc(checker: ^Checker, expr_id: ast.Expr_Id) -> Constant {
 			right = checker.constants[expr.right]
 		}
 		result := Constant{kind = .Not_Constant}
-		if left.kind == .Div_By_Zero || right.kind == .Div_By_Zero {
-			result = Constant{kind = .Div_By_Zero}
-		} else if left.kind == .Overflow || right.kind == .Overflow {
+		if left.kind == .Overflow || right.kind == .Overflow {
 			result = Constant{kind = .Overflow}
 		} else if left.kind == .Value && right.kind == .Value {
 			value: i128
 			overflow: bool
-			div_by_zero: bool
 			#partial switch expr.kind {
 			case .Sub: value, overflow = intrinsics.overflow_sub(left.value, right.value)
 			case .Mul: value, overflow = intrinsics.overflow_mul(left.value, right.value)
-			case .Div:
-				if right.value == 0 {
-					div_by_zero = true
-				} else {
-					value = left.value / right.value
-				}
 			case: value, overflow = intrinsics.overflow_add(left.value, right.value)
 			}
 			switch {
-			case div_by_zero: result = Constant{kind = .Div_By_Zero}
-			case overflow:    result = Constant{kind = .Overflow}
-			case:             result = Constant{kind = .Value, value = value}
+			case overflow: result = Constant{kind = .Overflow}
+			case:          result = Constant{kind = .Value, value = value}
 			}
 		}
 		checker.constants[frame.expr] = result
@@ -226,6 +217,8 @@ Ct_Error_Kind :: enum u8 {
 	Not_Comptime,
 	Overflow,
 	Div_By_Zero,
+	Non_Exact,
+	Integer_Division,
 	Quota,
 }
 
@@ -1105,7 +1098,8 @@ ct_eval_expr :: proc(
 		}
 		return ct_eval_unary(state, expr.kind, value, expr.span)
 	case .Add, .Sub, .Mul, .Div, .Eq, .Ne, .Lt, .Le, .Gt, .Ge:
-		left, flow, ok := ct_eval_expr(state, expr.left, types.INVALID, depth+1)
+		left_expected := expected if expr.kind == .Div && types.is_float(expected, checker.target) else types.INVALID
+		left, flow, ok := ct_eval_expr(state, expr.left, left_expected, depth+1)
 		if !ok || flow.kind != .Normal {
 			return INVALID_CT_VALUE, flow, ok
 		}
@@ -1858,6 +1852,12 @@ ct_eval_binary :: proc(state: ^Ct_State, op: ast.Expr_Kind, left_id, right_id: C
 			}
 			return ct_add_value(state, Ct_Value{kind=.Bool, type=types.BOOL, integer=1 if ok else 0}), ct_flow(.Normal), true
 		}
+		if op == .Div {
+			return INVALID_CT_VALUE, ct_flow(.Normal), ct_fail(
+				state, .Integer_Division, span,
+				"integer '/' is not allowed; use div_trunc, div_floor, div_exact, or div_ceil",
+			)
+		}
 		value: i128
 		overflow := false
 		#partial switch op {
@@ -1865,12 +1865,6 @@ ct_eval_binary :: proc(state: ^Ct_State, op: ast.Expr_Kind, left_id, right_id: C
 			value, overflow = intrinsics.overflow_sub(left.integer, right.integer)
 		case .Mul:
 			value, overflow = intrinsics.overflow_mul(left.integer, right.integer)
-		case .Div:
-			if right.integer == 0 {
-				state.error = .Div_By_Zero
-				return INVALID_CT_VALUE, ct_flow(.Normal), false
-			}
-			value = left.integer / right.integer
 		case:
 			value, overflow = intrinsics.overflow_add(left.integer, right.integer)
 		}
@@ -1887,6 +1881,137 @@ ct_eval_binary :: proc(state: ^Ct_State, op: ast.Expr_Kind, left_id, right_id: C
 	return INVALID_CT_VALUE, ct_flow(.Normal), ct_fail(state, .Not_Comptime, span, "comptime binary expression requires compatible operands")
 }
 
+ct_eval_division_builtin :: proc(
+	state: ^Ct_State,
+	kind: Division_Builtin,
+	left_id, right_id: Ct_Value_Id,
+	span: source.Span,
+) -> (Ct_Value_Id, Ct_Flow, bool) {
+	if left_id == INVALID_CT_VALUE || right_id == INVALID_CT_VALUE ||
+	   int(left_id) >= len(state.values) || int(right_id) >= len(state.values) {
+		return INVALID_CT_VALUE, ct_flow(.Normal), false
+	}
+	left := state.values[left_id]
+	right := state.values[right_id]
+	result_type := types.widest(left.type, right.type)
+	if !types.is_concrete_scalar(result_type) || types.is_bool(result_type) {
+		return INVALID_CT_VALUE, ct_flow(.Normal), ct_fail(
+			state, .Not_Comptime, span, "division builtins require compatible numeric operands",
+		)
+	}
+	left_id, left_ok := ct_coerce_value(state, left_id, result_type, span)
+	right_id, right_ok := ct_coerce_value(state, right_id, result_type, span)
+	if !left_ok || !right_ok {
+		return INVALID_CT_VALUE, ct_flow(.Normal), false
+	}
+	left = state.values[left_id]
+	right = state.values[right_id]
+	if left.kind == .Float && right.kind == .Float {
+		if right.float == 0 {
+			return INVALID_CT_VALUE, ct_flow(.Normal), ct_fail(state, .Div_By_Zero, span, "division builtin denominator is zero")
+		}
+		quotient := left.float / right.float
+		result := quotient
+		#partial switch kind {
+		case .Trunc: result = math.trunc(quotient)
+		case .Floor: result = math.floor(quotient)
+		case .Ceil:  result = math.ceil(quotient)
+		case .Exact:
+			result = math.trunc(quotient)
+			if result * right.float != left.float {
+				return INVALID_CT_VALUE, ct_flow(.Normal), ct_fail(state, .Non_Exact, span, "exact division has a remainder")
+			}
+		case .Rem, .Mod:
+			result = left.float - math.trunc(quotient) * right.float
+			if kind == .Mod && result != 0 && (result < 0) != (right.float < 0) {
+				result += right.float
+			}
+		}
+		if types.bits(result_type, state.checker.target) == 32 {
+			result = f64(f32(result))
+		}
+		return ct_add_value(state, Ct_Value{kind=.Float, type=result_type, float=result}), ct_flow(.Normal), true
+	}
+	if left.kind != .Integer || right.kind != .Integer {
+		return INVALID_CT_VALUE, ct_flow(.Normal), ct_fail(state, .Not_Comptime, span, "division builtins require compatible numeric operands")
+	}
+	if right.integer == 0 {
+		return INVALID_CT_VALUE, ct_flow(.Normal), ct_fail(state, .Div_By_Zero, span, "division builtin denominator is zero")
+	}
+	is_quotient := kind == .Trunc || kind == .Floor || kind == .Exact || kind == .Ceil
+	if is_quotient && types.is_signed(result_type, state.checker.target) {
+		minimum := -(i128(1) << u32(types.bits(result_type, state.checker.target)-1))
+		if left.integer == minimum && right.integer == -1 {
+			return INVALID_CT_VALUE, ct_flow(.Normal), ct_fail(state, .Overflow, span, "signed integer division overflow")
+		}
+	}
+	quotient := left.integer / right.integer
+	remainder := left.integer % right.integer
+	result := quotient
+	#partial switch kind {
+	case .Floor:
+		if remainder != 0 && (left.integer < 0) != (right.integer < 0) {
+			result -= 1
+		}
+	case .Ceil:
+		if remainder != 0 && (left.integer < 0) == (right.integer < 0) {
+			result += 1
+		}
+	case .Exact:
+		if remainder != 0 {
+			return INVALID_CT_VALUE, ct_flow(.Normal), ct_fail(state, .Non_Exact, span, "exact division has a remainder")
+		}
+	case .Rem: result = remainder
+	case .Mod:
+		result = remainder
+		if result != 0 && (result < 0) != (right.integer < 0) {
+			result += right.integer
+		}
+	case:
+	}
+	return ct_add_value(state, Ct_Value{kind=.Integer, type=result_type, integer=result}), ct_flow(.Normal), true
+}
+
+ct_eval_division_call :: proc(
+	state: ^Ct_State,
+	expr: ast.Expr,
+	kind: Division_Builtin,
+	expected: types.Type,
+	depth: int,
+) -> (Ct_Value_Id, Ct_Flow, bool) {
+	checker := state.checker
+	if len(expr.args) != 2 {
+		return INVALID_CT_VALUE, ct_flow(.Normal), ct_failf(
+			state, .Not_Comptime, expr.span, "%s expects 2 arguments, got %d",
+			symbol_text(checker, expr.name), len(expr.args),
+		)
+	}
+	hint := expected if types.is_concrete_scalar(expected) && !types.is_bool(expected) else types.INVALID
+	left_const := is_numeric_constant_expr(checker, expr.args[0])
+	right_const := is_numeric_constant_expr(checker, expr.args[1])
+	left, right := INVALID_CT_VALUE, INVALID_CT_VALUE
+	flow := ct_flow(.Normal)
+	ok := false
+	if left_const && !right_const && !types.is_valid(hint) {
+		right, flow, ok = ct_eval_expr(state, expr.args[1], types.INVALID, depth+1)
+		if !ok || flow.kind != .Normal {
+			return INVALID_CT_VALUE, flow, ok
+		}
+		left, flow, ok = ct_eval_expr(state, expr.args[0], state.values[right].type, depth+1)
+	} else {
+		left, flow, ok = ct_eval_expr(state, expr.args[0], hint, depth+1)
+		if !ok || flow.kind != .Normal {
+			return INVALID_CT_VALUE, flow, ok
+		}
+		right_hint := hint if types.is_valid(hint) else state.values[left].type
+		right, flow, ok = ct_eval_expr(state, expr.args[1], right_hint, depth+1)
+	}
+	if !ok || flow.kind != .Normal {
+		return INVALID_CT_VALUE, flow, ok
+	}
+	return ct_eval_division_builtin(state, kind, left, right, expr.span)
+}
+
 ct_scalar_cast :: proc(state: ^Ct_State, id: Ct_Value_Id, target: types.Type, span: source.Span) -> (Ct_Value_Id, Ct_Flow, bool) {
 	if id == INVALID_CT_VALUE || int(id) >= len(state.values) {
 		return INVALID_CT_VALUE, ct_flow(.Normal), false
@@ -1943,6 +2068,9 @@ ct_eval_call_expr :: proc(state: ^Ct_State, expr: ast.Expr, expected: types.Type
 		result_type := types.USIZE if builtin == .Size_Of || builtin == .Align_Of else target
 		return ct_add_value(state, Ct_Value{kind=.Integer, type=result_type, integer=type_builtin_value(checker, builtin, target)}), ct_flow(.Normal), true
 	}
+	if builtin := division_builtin_call(checker, expr); builtin != .None {
+		return ct_eval_division_call(state, expr, builtin, expected, depth+1)
+	}
 	target_pkg, available := expr_package(checker, expr, state.pkg, state.file, false)
 	if !available {
 		return INVALID_CT_VALUE, ct_flow(.Normal), ct_fail(state, .Not_Comptime, expr.span, "unavailable function package")
@@ -2899,6 +3027,10 @@ eval_integer_constant_in_context :: proc(
 			return Constant{kind=.Overflow}
 		case .Div_By_Zero:
 			return Constant{kind=.Div_By_Zero}
+		case .Non_Exact:
+			return Constant{kind=.Non_Exact}
+		case .Integer_Division:
+			return Constant{kind=.Integer_Division}
 		}
 		return Constant{kind=.Not_Constant}
 	}
@@ -2927,6 +3059,10 @@ eval_comptime_statements :: proc(
 			return Constant{kind=.Overflow}, false, false
 		case .Div_By_Zero:
 			return Constant{kind=.Div_By_Zero}, false, false
+		case .Non_Exact:
+			return Constant{kind=.Non_Exact}, false, false
+		case .Integer_Division:
+			return Constant{kind=.Integer_Division}, false, false
 		}
 		return Constant{kind=.Not_Constant}, false, false
 	}
@@ -2958,6 +3094,10 @@ eval_comptime_call :: proc(
 			return Constant{kind=.Overflow}
 		case .Div_By_Zero:
 			return Constant{kind=.Div_By_Zero}
+		case .Non_Exact:
+			return Constant{kind=.Non_Exact}
+		case .Integer_Division:
+			return Constant{kind=.Integer_Division}
 		}
 		return Constant{kind=.Not_Constant}
 	}
@@ -2996,7 +3136,7 @@ infer_comptime_expr_type :: proc(
 		}
 	}
 	if !ok || flow.kind != .Normal || value == INVALID_CT_VALUE || int(value) >= len(state.values) {
-		if state.error == .Overflow || state.error == .Div_By_Zero {
+		if state.error == .Overflow || state.error == .Div_By_Zero || state.error == .Non_Exact || state.error == .Integer_Division {
 			return types.I64
 		}
 		return types.INVALID
@@ -3038,6 +3178,12 @@ build_comptime_expr :: proc(
 	if state.error == .Overflow {
 		return build_constant_expr(checker, expr, Constant{kind=.Overflow}, expected)
 	}
+	if state.error == .Non_Exact {
+		return build_constant_expr(checker, expr, Constant{kind=.Non_Exact}, expected)
+	}
+	if state.error == .Integer_Division {
+		return build_constant_expr(checker, expr, Constant{kind=.Integer_Division}, expected)
+	}
 	diagnostic := state.diagnostic
 	if diagnostic == source.INVALID_DIAGNOSTIC {
 		diagnostic = source.add(checker.diagnostics, expr.span, "expression cannot be evaluated at comptime")
diff --git a/compiler/hir/hir.odin b/compiler/hir/hir.odin
index b99e0a9..6515449 100644
--- a/compiler/hir/hir.odin
+++ b/compiler/hir/hir.odin
@@ -113,6 +113,12 @@ Expr_Kind :: enum u8 {
 	Sub,
 	Mul,
 	Div,
+	Div_Trunc,
+	Div_Floor,
+	Div_Exact,
+	Div_Ceil,
+	Rem,
+	Mod,
 	Pointer_Add,
 	Eq,
 	Ne,
diff --git a/compiler/ir/ir.odin b/compiler/ir/ir.odin
index b3cf6f3..6ac7ef4 100644
--- a/compiler/ir/ir.odin
+++ b/compiler/ir/ir.odin
@@ -109,6 +109,12 @@ Opcode :: enum u8 {
 	Sub_Checked,
 	Mul_Checked,
 	Div_Checked,
+	Div_Trunc_Checked,
+	Div_Floor_Checked,
+	Div_Exact_Checked,
+	Div_Ceil_Checked,
+	Rem_Checked,
+	Mod_Checked,
 	Pointer_Add,
 	Not,
 	Compare,
diff --git a/compiler/llvm/llvm.odin b/compiler/llvm/llvm.odin
index 6edbcfd..c97a487 100644
--- a/compiler/llvm/llvm.odin
+++ b/compiler/llvm/llvm.odin
@@ -257,7 +257,10 @@ valid_value :: proc(
 	     .Fallible_Error, .Extract, .Select, .Unwrap,
 	     .Optional_Is_Some, .Optional_Value, .Orelse,
 	     .Widen, .Sum_Widen, .C_Coerce, .C_Vararg_Promote, .Retype, .Scalar_Cast, .Pointer_Cast, .Weaken_Pointer, .Weaken_Slice, .Decay_Array_Pointer,
-	     .Neg_Checked, .Add_Checked, .Sub_Checked, .Mul_Checked, .Div_Checked, .Pointer_Add, .Not, .Compare, .Call:
+	     .Neg_Checked, .Add_Checked, .Sub_Checked, .Mul_Checked, .Div_Checked,
+	     .Div_Trunc_Checked, .Div_Floor_Checked, .Div_Exact_Checked, .Div_Ceil_Checked,
+	     .Rem_Checked, .Mod_Checked,
+	     .Pointer_Add, .Not, .Compare, .Call:
 		return true
 	case .Address_Global, .Alloca, .Index_Address, .Field_Address, .Orelse_Begin,
 	     .Store, .Fill, .Trap, .Label, .Br, .Cond_Br, .Return, .Return_Void:
@@ -582,9 +585,7 @@ emit_checked_arithmetic :: proc(
 	fmt.sbprintf(&emitter.builder, "  unreachable\noverflow_continue%d:\n", instruction_index)
 }
 
-// emit_checked_division emits a trapping integer division guarding divide-by-zero
-// and signed `INT_MIN / -1` overflow, or a plain floating-point division.
-emit_checked_division :: proc(
+emit_division_zero_guard :: proc(
 	emitter: ^Emitter,
 	instructions: []ir.Instruction,
 	instruction_index: int,
@@ -592,56 +593,225 @@ emit_checked_division :: proc(
 ) {
 	type_name := llvm_type(instruction.type, &emitter.module.types)
 	if types.is_float(instruction.type, emitter.module.target) {
-		fmt.sbprintf(&emitter.builder, "  %%v%d = fdiv %s ", instruction_index, type_name)
-		write_operand(&emitter.builder, instructions, instruction.a, instruction.type, &emitter.module.types)
-		strings.write_string(&emitter.builder, ", ")
+		fmt.sbprintf(&emitter.builder, "  %%divzero%d = fcmp oeq %s ", instruction_index, type_name)
 		write_operand(&emitter.builder, instructions, instruction.b, instruction.type, &emitter.module.types)
-		strings.write_string(&emitter.builder, "\n")
-		return
+		strings.write_string(&emitter.builder, ", 0.000000e+00\n")
+	} else {
+		fmt.sbprintf(&emitter.builder, "  %%divzero%d = icmp eq %s ", instruction_index, type_name)
+		write_operand(&emitter.builder, instructions, instruction.b, instruction.type, &emitter.module.types)
+		strings.write_string(&emitter.builder, ", 0\n")
 	}
-	signed := !types.is_unsigned(instruction.type, emitter.module.target)
-	fmt.sbprintf(&emitter.builder, "  %%divzero%d = icmp eq %s ", instruction_index, type_name)
-	write_operand(&emitter.builder, instructions, instruction.b, instruction.type, &emitter.module.types)
-	strings.write_string(&emitter.builder, ", 0\n")
 	fmt.sbprintf(
 		&emitter.builder,
-		"  br i1 %%divzero%d, label %%divzero_trap%d, label %%divzero_ok%d\n",
+		"  br i1 %%divzero%d, label %%divzero_trap%d, label %%divzero_ok%d\ndivzero_trap%d:\n",
+		instruction_index,
 		instruction_index,
 		instruction_index,
 		instruction_index,
 	)
-	fmt.sbprintf(&emitter.builder, "divzero_trap%d:\n", instruction_index)
-	zero_message := diagnostic_message(emitter, source.INVALID_DIAGNOSTIC, instruction.span, "integer division by zero")
-	emit_trap_call(emitter, zero_message)
+	message := diagnostic_message(emitter, source.INVALID_DIAGNOSTIC, instruction.span, "division builtin denominator is zero")
+	emit_trap_call(emitter, message)
 	fmt.sbprintf(&emitter.builder, "  unreachable\ndivzero_ok%d:\n", instruction_index)
-	if signed {
-		min_value := -(i128(1) << u32(types.bits(instruction.type, emitter.module.target) - 1))
-		fmt.sbprintf(&emitter.builder, "  %%divminlo%d = icmp eq %s ", instruction_index, type_name)
+}
+
+emit_division_overflow_guard :: proc(
+	emitter: ^Emitter,
+	instructions: []ir.Instruction,
+	instruction_index: int,
+	instruction: ir.Instruction,
+) {
+	type_name := llvm_type(instruction.type, &emitter.module.types)
+	min_value := -(i128(1) << u32(types.bits(instruction.type, emitter.module.target)-1))
+	fmt.sbprintf(&emitter.builder, "  %%divminlo%d = icmp eq %s ", instruction_index, type_name)
+	write_operand(&emitter.builder, instructions, instruction.a, instruction.type, &emitter.module.types)
+	fmt.sbprintf(&emitter.builder, ", %d\n  %%divminhi%d = icmp eq %s ", min_value, instruction_index, type_name)
+	write_operand(&emitter.builder, instructions, instruction.b, instruction.type, &emitter.module.types)
+	fmt.sbprintf(
+		&emitter.builder,
+		", -1\n  %%divovf%d = and i1 %%divminlo%d, %%divminhi%d\n  br i1 %%divovf%d, label %%divovf_trap%d, label %%divovf_ok%d\ndivovf_trap%d:\n",
+		instruction_index,
+		instruction_index,
+		instruction_index,
+		instruction_index,
+		instruction_index,
+		instruction_index,
+		instruction_index,
+	)
+	message := diagnostic_message(emitter, source.INVALID_DIAGNOSTIC, instruction.span, "signed integer division overflow")
+	emit_trap_call(emitter, message)
+	fmt.sbprintf(&emitter.builder, "  unreachable\ndivovf_ok%d:\n", instruction_index)
+}
+
+emit_float_division_builtin :: proc(
+	emitter: ^Emitter,
+	instructions: []ir.Instruction,
+	instruction_index: int,
+	instruction: ir.Instruction,
+) {
+	type_name := llvm_type(instruction.type, &emitter.module.types)
+	suffix := "f32" if types.bits(instruction.type, emitter.module.target) == 32 else "f64"
+
+	if instruction.op == .Rem_Checked || instruction.op == .Mod_Checked {
+		name := fmt.tprintf("%%v%d", instruction_index) if instruction.op == .Rem_Checked else fmt.tprintf("%%rawrem%d", instruction_index)
+		fmt.sbprintf(&emitter.builder, "  %s = frem %s ", name, type_name)
 		write_operand(&emitter.builder, instructions, instruction.a, instruction.type, &emitter.module.types)
-		fmt.sbprintf(&emitter.builder, ", %d\n", min_value)
-		fmt.sbprintf(&emitter.builder, "  %%divminhi%d = icmp eq %s ", instruction_index, type_name)
+		strings.write_string(&emitter.builder, ", ")
 		write_operand(&emitter.builder, instructions, instruction.b, instruction.type, &emitter.module.types)
-		strings.write_string(&emitter.builder, ", -1\n")
-		fmt.sbprintf(&emitter.builder, "  %%divovf%d = and i1 %%divminlo%d, %%divminhi%d\n", instruction_index, instruction_index, instruction_index)
-		fmt.sbprintf(
-			&emitter.builder,
-			"  br i1 %%divovf%d, label %%divovf_trap%d, label %%divovf_ok%d\n",
-			instruction_index,
-			instruction_index,
-			instruction_index,
-		)
-		fmt.sbprintf(&emitter.builder, "divovf_trap%d:\n", instruction_index)
-		ovf_message := diagnostic_message(emitter, source.INVALID_DIAGNOSTIC, instruction.span, "signed integer division overflow")
-		emit_trap_call(emitter, ovf_message)
-		fmt.sbprintf(&emitter.builder, "  unreachable\ndivovf_ok%d:\n", instruction_index)
-		fmt.sbprintf(&emitter.builder, "  %%v%d = sdiv %s ", instruction_index, type_name)
-	} else {
-		fmt.sbprintf(&emitter.builder, "  %%v%d = udiv %s ", instruction_index, type_name)
+		strings.write_string(&emitter.builder, "\n")
+		if instruction.op == .Rem_Checked {
+			return
+		}
+		fmt.sbprintf(&emitter.builder, "  %%remnonzero%d = fcmp one %s %%rawrem%d, 0.000000e+00\n", instruction_index, type_name, instruction_index)
+		fmt.sbprintf(&emitter.builder, "  %%remsign%d = fcmp olt %s %%rawrem%d, 0.000000e+00\n", instruction_index, type_name, instruction_index)
+		fmt.sbprintf(&emitter.builder, "  %%denomsign%d = fcmp olt %s ", instruction_index, type_name)
+		write_operand(&emitter.builder, instructions, instruction.b, instruction.type, &emitter.module.types)
+		strings.write_string(&emitter.builder, ", 0.000000e+00\n")
+		fmt.sbprintf(&emitter.builder, "  %%signsdiffer%d = xor i1 %%remsign%d, %%denomsign%d\n", instruction_index, instruction_index, instruction_index)
+		fmt.sbprintf(&emitter.builder, "  %%modadjust%d = and i1 %%remnonzero%d, %%signsdiffer%d\n", instruction_index, instruction_index, instruction_index)
+		fmt.sbprintf(&emitter.builder, "  %%adjustedrem%d = fadd %s %%rawrem%d, ", instruction_index, type_name, instruction_index)
+		write_operand(&emitter.builder, instructions, instruction.b, instruction.type, &emitter.module.types)
+		fmt.sbprintf(&emitter.builder, "\n  %%v%d = select i1 %%modadjust%d, %s %%adjustedrem%d, %s %%rawrem%d\n", instruction_index, instruction_index, type_name, instruction_index, type_name, instruction_index)
+		return
 	}
+
+	fmt.sbprintf(&emitter.builder, "  %%divq%d = fdiv %s ", instruction_index, type_name)
 	write_operand(&emitter.builder, instructions, instruction.a, instruction.type, &emitter.module.types)
 	strings.write_string(&emitter.builder, ", ")
 	write_operand(&emitter.builder, instructions, instruction.b, instruction.type, &emitter.module.types)
 	strings.write_string(&emitter.builder, "\n")
+	intrinsic := "trunc"
+	if instruction.op == .Div_Floor_Checked {
+		intrinsic = "floor"
+	} else if instruction.op == .Div_Ceil_Checked {
+		intrinsic = "ceil"
+	}
+	fmt.sbprintf(&emitter.builder, "  %%v%d = call %s @llvm.%s.%s(%s %%divq%d)\n", instruction_index, type_name, intrinsic, suffix, type_name, instruction_index)
+	if instruction.op != .Div_Exact_Checked {
+		return
+	}
+	fmt.sbprintf(&emitter.builder, "  %%exactprod%d = fmul %s %%v%d, ", instruction_index, type_name, instruction_index)
+	write_operand(&emitter.builder, instructions, instruction.b, instruction.type, &emitter.module.types)
+	fmt.sbprintf(&emitter.builder, "\n  %%exact%d = fcmp oeq %s %%exactprod%d, ", instruction_index, type_name, instruction_index)
+	write_operand(&emitter.builder, instructions, instruction.a, instruction.type, &emitter.module.types)
+	fmt.sbprintf(&emitter.builder, "\n  br i1 %%exact%d, label %%exact_ok%d, label %%exact_trap%d\nexact_trap%d:\n", instruction_index, instruction_index, instruction_index, instruction_index)
+	message := diagnostic_message(emitter, source.INVALID_DIAGNOSTIC, instruction.span, "exact division has a remainder")
+	emit_trap_call(emitter, message)
+	fmt.sbprintf(&emitter.builder, "  unreachable\nexact_ok%d:\n", instruction_index)
+}
+
+emit_integer_remainder_builtin :: proc(
+	emitter: ^Emitter,
+	instructions: []ir.Instruction,
+	instruction_index: int,
+	instruction: ir.Instruction,
+) {
+	type_name := llvm_type(instruction.type, &emitter.module.types)
+	signed := types.is_signed(instruction.type, emitter.module.target)
+	raw_name := fmt.tprintf("%%v%d", instruction_index) if instruction.op == .Rem_Checked || !signed else fmt.tprintf("%%rawrem%d", instruction_index)
+	if !signed {
+		fmt.sbprintf(&emitter.builder, "  %s = urem %s ", raw_name, type_name)
+		write_operand(&emitter.builder, instructions, instruction.a, instruction.type, &emitter.module.types)
+		strings.write_string(&emitter.builder, ", ")
+		write_operand(&emitter.builder, instructions, instruction.b, instruction.type, &emitter.module.types)
+		strings.write_string(&emitter.builder, "\n")
+	} else {
+		min_value := -(i128(1) << u32(types.bits(instruction.type, emitter.module.target)-1))
+		fmt.sbprintf(&emitter.builder, "  %%remminlo%d = icmp eq %s ", instruction_index, type_name)
+		write_operand(&emitter.builder, instructions, instruction.a, instruction.type, &emitter.module.types)
+		fmt.sbprintf(&emitter.builder, ", %d\n  %%remminhi%d = icmp eq %s ", min_value, instruction_index, type_name)
+		write_operand(&emitter.builder, instructions, instruction.b, instruction.type, &emitter.module.types)
+		fmt.sbprintf(&emitter.builder, ", -1\n  %%remspecial%d = and i1 %%remminlo%d, %%remminhi%d\n", instruction_index, instruction_index, instruction_index)
+		fmt.sbprintf(&emitter.builder, "  br i1 %%remspecial%d, label %%rem_special%d, label %%rem_normal%d\nrem_special%d:\n  br label %%rem_join%d\nrem_normal%d:\n", instruction_index, instruction_index, instruction_index, instruction_index, instruction_index, instruction_index)
+		fmt.sbprintf(&emitter.builder, "  %%remnormal%d = srem %s ", instruction_index, type_name)
+		write_operand(&emitter.builder, instructions, instruction.a, instruction.type, &emitter.module.types)
+		strings.write_string(&emitter.builder, ", ")
+		write_operand(&emitter.builder, instructions, instruction.b, instruction.type, &emitter.module.types)
+		fmt.sbprintf(&emitter.builder, "\n  br label %%rem_join%d\nrem_join%d:\n  %s = phi %s [ 0, %%rem_special%d ], [ %%remnormal%d, %%rem_normal%d ]\n", instruction_index, instruction_index, raw_name, type_name, instruction_index, instruction_index, instruction_index)
+	}
+	if instruction.op == .Rem_Checked || !signed {
+		return
+	}
+	fmt.sbprintf(&emitter.builder, "  %%remnonzero%d = icmp ne %s %%rawrem%d, 0\n", instruction_index, type_name, instruction_index)
+	fmt.sbprintf(&emitter.builder, "  %%remsign%d = icmp slt %s %%rawrem%d, 0\n", instruction_index, type_name, instruction_index)
+	fmt.sbprintf(&emitter.builder, "  %%denomsign%d = icmp slt %s ", instruction_index, type_name)
+	write_operand(&emitter.builder, instructions, instruction.b, instruction.type, &emitter.module.types)
+	fmt.sbprintf(&emitter.builder, ", 0\n  %%signsdiffer%d = xor i1 %%remsign%d, %%denomsign%d\n", instruction_index, instruction_index, instruction_index)
+	fmt.sbprintf(&emitter.builder, "  %%modadjust%d = and i1 %%remnonzero%d, %%signsdiffer%d\n  %%adjustedrem%d = add %s %%rawrem%d, ", instruction_index, instruction_index, instruction_index, instruction_index, type_name, instruction_index)
+	write_operand(&emitter.builder, instructions, instruction.b, instruction.type, &emitter.module.types)
+	fmt.sbprintf(&emitter.builder, "\n  %%v%d = select i1 %%modadjust%d, %s %%adjustedrem%d, %s %%rawrem%d\n", instruction_index, instruction_index, type_name, instruction_index, type_name, instruction_index)
+}
+
+emit_integer_quotient_builtin :: proc(
+	emitter: ^Emitter,
+	instructions: []ir.Instruction,
+	instruction_index: int,
+	instruction: ir.Instruction,
+) {
+	type_name := llvm_type(instruction.type, &emitter.module.types)
+	signed := types.is_signed(instruction.type, emitter.module.target)
+	operation := "sdiv" if signed else "udiv"
+	name := fmt.tprintf("%%v%d", instruction_index) if instruction.op == .Div_Trunc_Checked else fmt.tprintf("%%divq%d", instruction_index)
+	fmt.sbprintf(&emitter.builder, "  %s = %s %s ", name, operation, type_name)
+	write_operand(&emitter.builder, instructions, instruction.a, instruction.type, &emitter.module.types)
+	strings.write_string(&emitter.builder, ", ")
+	write_operand(&emitter.builder, instructions, instruction.b, instruction.type, &emitter.module.types)
+	strings.write_string(&emitter.builder, "\n")
+	if instruction.op == .Div_Trunc_Checked {
+		return
+	}
+	fmt.sbprintf(&emitter.builder, "  %%divprod%d = mul %s %%divq%d, ", instruction_index, type_name, instruction_index)
+	write_operand(&emitter.builder, instructions, instruction.b, instruction.type, &emitter.module.types)
+	fmt.sbprintf(&emitter.builder, "\n  %%divrem%d = sub %s ", instruction_index, type_name)
+	write_operand(&emitter.builder, instructions, instruction.a, instruction.type, &emitter.module.types)
+	fmt.sbprintf(&emitter.builder, ", %%divprod%d\n", instruction_index)
+	if instruction.op == .Div_Exact_Checked {
+		fmt.sbprintf(&emitter.builder, "  %%exact%d = icmp eq %s %%divrem%d, 0\n  br i1 %%exact%d, label %%exact_ok%d, label %%exact_trap%d\nexact_trap%d:\n", instruction_index, type_name, instruction_index, instruction_index, instruction_index, instruction_index, instruction_index)
+		message := diagnostic_message(emitter, source.INVALID_DIAGNOSTIC, instruction.span, "exact division has a remainder")
+		emit_trap_call(emitter, message)
+		fmt.sbprintf(&emitter.builder, "  unreachable\nexact_ok%d:\n  %%v%d = add %s %%divq%d, 0\n", instruction_index, instruction_index, type_name, instruction_index)
+		return
+	}
+	fmt.sbprintf(&emitter.builder, "  %%remnonzero%d = icmp ne %s %%divrem%d, 0\n", instruction_index, type_name, instruction_index)
+	if signed {
+		fmt.sbprintf(&emitter.builder, "  %%numsign%d = icmp slt %s ", instruction_index, type_name)
+		write_operand(&emitter.builder, instructions, instruction.a, instruction.type, &emitter.module.types)
+		fmt.sbprintf(&emitter.builder, ", 0\n  %%denomsign%d = icmp slt %s ", instruction_index, type_name)
+		write_operand(&emitter.builder, instructions, instruction.b, instruction.type, &emitter.module.types)
+		fmt.sbprintf(&emitter.builder, ", 0\n  %%signsdiffer%d = xor i1 %%numsign%d, %%denomsign%d\n", instruction_index, instruction_index, instruction_index)
+		predicate := fmt.tprintf("%%signsdiffer%d", instruction_index)
+		if instruction.op == .Div_Ceil_Checked {
+			fmt.sbprintf(&emitter.builder, "  %%signssame%d = xor i1 %%signsdiffer%d, true\n", instruction_index, instruction_index)
+			predicate = fmt.tprintf("%%signssame%d", instruction_index)
+		}
+		fmt.sbprintf(&emitter.builder, "  %%divadjust%d = and i1 %%remnonzero%d, %s\n", instruction_index, instruction_index, predicate)
+	} else {
+		fmt.sbprintf(&emitter.builder, "  %%divadjust%d = and i1 %%remnonzero%d, true\n", instruction_index, instruction_index)
+	}
+	adjustment := "sub" if instruction.op == .Div_Floor_Checked else "add"
+	fmt.sbprintf(&emitter.builder, "  %%adjustedq%d = %s %s %%divq%d, 1\n  %%v%d = select i1 %%divadjust%d, %s %%adjustedq%d, %s %%divq%d\n", instruction_index, adjustment, type_name, instruction_index, instruction_index, instruction_index, type_name, instruction_index, type_name, instruction_index)
+}
+
+emit_division_builtin :: proc(
+	emitter: ^Emitter,
+	instructions: []ir.Instruction,
+	instruction_index: int,
+	instruction: ir.Instruction,
+) {
+	emit_division_zero_guard(emitter, instructions, instruction_index, instruction)
+	if types.is_float(instruction.type, emitter.module.target) {
+		emit_float_division_builtin(emitter, instructions, instruction_index, instruction)
+		return
+	}
+	quotient := instruction.op == .Div_Trunc_Checked || instruction.op == .Div_Floor_Checked ||
+	            instruction.op == .Div_Exact_Checked || instruction.op == .Div_Ceil_Checked
+	if quotient && types.is_signed(instruction.type, emitter.module.target) {
+		emit_division_overflow_guard(emitter, instructions, instruction_index, instruction)
+	}
+	if quotient {
+		emit_integer_quotient_builtin(emitter, instructions, instruction_index, instruction)
+	} else {
+		emit_integer_remainder_builtin(emitter, instructions, instruction_index, instruction)
+	}
 }
 
 emit_instruction_stream :: proc(
@@ -1609,11 +1779,22 @@ emit_instruction_stream :: proc(
 			emit_checked_arithmetic(emitter, instructions, instruction_index, instruction, "mul", "fmul", "integer multiplication overflow")
 		case .Div_Checked:
 			if !valid_value(instructions, instruction.a, instruction.type, &emitter.module.types) ||
-			   !valid_value(instructions, instruction.b, instruction.type, &emitter.module.types) {
+			   !valid_value(instructions, instruction.b, instruction.type, &emitter.module.types) ||
+			   !types.is_float(instruction.type, emitter.module.target) {
 				emit_recovery_value(emitter, instruction_index, instruction, "invalid division operand")
 				continue
 			}
-			emit_checked_division(emitter, instructions, instruction_index, instruction)
+			emit_checked_arithmetic(emitter, instructions, instruction_index, instruction, "div", "fdiv", "")
+		case .Div_Trunc_Checked, .Div_Floor_Checked, .Div_Exact_Checked, .Div_Ceil_Checked,
+		     .Rem_Checked, .Mod_Checked:
+			if !valid_value(instructions, instruction.a, instruction.type, &emitter.module.types) ||
+			   !valid_value(instructions, instruction.b, instruction.type, &emitter.module.types) ||
+			   (!types.is_concrete_integer(instruction.type) &&
+			    !types.is_float(instruction.type, emitter.module.target)) {
+				emit_recovery_value(emitter, instruction_index, instruction, "invalid division builtin operands")
+				continue
+			}
+			emit_division_builtin(emitter, instructions, instruction_index, instruction)
 		case .Pointer_Add:
 			result_item, result_ok := types.node(&emitter.module.types, instruction.type)
 			base_type := instructions[instruction.a].type if valid_instruction(instructions, instruction.a) else types.INVALID
@@ -2283,6 +2464,7 @@ emit_messages :: proc(emitter: ^Emitter) {
 
 emit_declarations :: proc(emitter: ^Emitter) {
 	strings.write_string(&emitter.builder, "declare i64 @write(i32, ptr, i64)\ndeclare void @llvm.trap()\ndeclare void @llvm.memcpy.p0.p0.i64(ptr, ptr, i64, i1 immarg)\ndeclare void @llvm.memset.p0.i64(ptr, i8, i64, i1 immarg)\n")
+	strings.write_string(&emitter.builder, "declare float @llvm.trunc.f32(float)\ndeclare double @llvm.trunc.f64(double)\ndeclare float @llvm.floor.f32(float)\ndeclare double @llvm.floor.f64(double)\ndeclare float @llvm.ceil.f32(float)\ndeclare double @llvm.ceil.f64(double)\n")
 	widths := [?]int{8, 16, 32, 64}
 	overflow_intrinsics := [?]string{"sadd", "uadd", "ssub", "usub", "smul", "umul"}
 	for bits in widths {
diff --git a/compiler/lower/lower.odin b/compiler/lower/lower.odin
index 4fa15ce..fc7d965 100644
--- a/compiler/lower/lower.odin
+++ b/compiler/lower/lower.odin
@@ -681,7 +681,8 @@ lower_expr :: proc(state: ^State, expr_id: hir.Expr_Id) -> ir.Instruction_Id {
 			case .Negate:
 				stack[frame_index].stage = 5
 				append(&stack, Lower_Expr_Frame{expr=expr.left})
-			case .Add, .Sub, .Mul, .Div, .Pointer_Add:
+			case .Add, .Sub, .Mul, .Div, .Div_Trunc, .Div_Floor, .Div_Exact, .Div_Ceil,
+			     .Rem, .Mod, .Pointer_Add:
 				stack[frame_index].stage = 2
 				append(&stack, Lower_Expr_Frame{expr=expr.left})
 			case .Call:
@@ -760,6 +761,12 @@ lower_expr :: proc(state: ^State, expr_id: hir.Expr_Id) -> ir.Instruction_Id {
 			case .Sub:         op = .Sub_Checked
 			case .Mul:         op = .Mul_Checked
 			case .Div:         op = .Div_Checked
+			case .Div_Trunc:   op = .Div_Trunc_Checked
+			case .Div_Floor:   op = .Div_Floor_Checked
+			case .Div_Exact:   op = .Div_Exact_Checked
+			case .Div_Ceil:    op = .Div_Ceil_Checked
+			case .Rem:         op = .Rem_Checked
+			case .Mod:         op = .Mod_Checked
 			case .Pointer_Add: op = .Pointer_Add
 			}
 			last = append_instruction(state, ir.Instruction{
diff --git a/compiler_tests.odin b/compiler_tests.odin
index 5e687f0..7e1490b 100644
--- a/compiler_tests.odin
+++ b/compiler_tests.odin
@@ -6246,7 +6246,7 @@ main func() void {}
 
 @(test)
 constant_division_by_zero_has_a_precise_diagnostic :: proc(t: ^testing.T) {
-	text := `value :: 5 / 0
+	text := `value :: div_trunc(5, 0)
 main func() void {}
 `
 	source_file := source.Source{path="test.bro", text=text}
@@ -6265,7 +6265,7 @@ main func() void {}
 	found_overflow := false
 	for diagnostic in diagnostics.items {
 		found_division_by_zero = found_division_by_zero ||
-		                         strings.contains(diagnostic.message, "division by zero in constant expression")
+		                         strings.contains(diagnostic.message, "division builtin denominator is zero")
 		found_overflow = found_overflow ||
 		                 strings.contains(diagnostic.message, "integer constant expression exceeds signed i64 range")
 	}
@@ -6404,11 +6404,19 @@ malformed_hir_references_lower_to_valid_trapped_llvm :: proc(t: ^testing.T) {
 malformed_ir_emits_traps_and_typed_sentinels :: proc(t: ^testing.T) {
 	module := ir.init_module()
 	defer ir.destroy_module(&module)
-	instructions := make([]ir.Instruction, 4)
+	instructions := make([]ir.Instruction, 11)
 	instructions[0] = ir.Instruction{op=.Store, type=types.I8, a=ir.INVALID_INSTRUCTION, b=ir.INVALID_INSTRUCTION, diagnostic=source.INVALID_DIAGNOSTIC}
 	instructions[1] = ir.Instruction{op=.Add_Checked, type=types.I32, a=ir.INVALID_INSTRUCTION, b=ir.INVALID_INSTRUCTION, diagnostic=source.INVALID_DIAGNOSTIC}
 	instructions[2] = ir.Instruction{op=.Neg_Checked, type=types.I16, a=ir.INVALID_INSTRUCTION, b=ir.INVALID_INSTRUCTION, diagnostic=source.INVALID_DIAGNOSTIC}
-	instructions[3] = ir.Instruction{op=.Return, type=types.I32, a=ir.Instruction_Id(1), b=ir.INVALID_INSTRUCTION, diagnostic=source.INVALID_DIAGNOSTIC}
+	division_ops := [?]ir.Opcode{
+		.Div_Checked,
+		.Div_Trunc_Checked, .Div_Floor_Checked, .Div_Exact_Checked, .Div_Ceil_Checked,
+		.Rem_Checked, .Mod_Checked,
+	}
+	for op, index in division_ops {
+		instructions[3+index] = ir.Instruction{op=op, type=types.I32, a=ir.INVALID_INSTRUCTION, b=ir.INVALID_INSTRUCTION, diagnostic=source.INVALID_DIAGNOSTIC}
+	}
+	instructions[10] = ir.Instruction{op=.Return, type=types.I32, a=ir.Instruction_Id(1), b=ir.INVALID_INSTRUCTION, diagnostic=source.INVALID_DIAGNOSTIC}
 	append(&module.functions, ir.Function{
 		link_name=strings.clone("main"),
 		calling_convention=.C,
@@ -6430,6 +6438,9 @@ malformed_ir_emits_traps_and_typed_sentinels :: proc(t: ^testing.T) {
 	testing.expect(t, strings.contains(text, "%v0 = add i8 0, -86"))
 	testing.expect(t, strings.contains(text, "%v1 = add i32 0, -1431655766"))
 	testing.expect(t, strings.contains(text, "%v2 = add i16 0, -21846"))
+	for index in 3..=9 {
+		testing.expect(t, strings.contains(text, fmt.tprintf("%%v%d = add i32 0, -1431655766", index)))
+	}
 	testing.expect(t, strings.contains(text, "@bro.trap(ptr %message, i64 %length) noreturn"))
 	testing.expect(t, !strings.contains(text, "%v-1"))
 	llvm_path := "/tmp/brolang-test-malformed-recovery.ll"
@@ -9200,12 +9211,12 @@ compound_assignment_preserves_checked_numeric_operations :: proc(t: ^testing.T)
 	signed += 6
 	signed -= 2
 	signed *= 3
-	signed /= 4
+	signed = div_trunc(signed, 4)
 	unsigned u32 = 24
 	unsigned += 6
 	unsigned -= 2
 	unsigned *= 3
-	unsigned /= 4
+	unsigned = div_trunc(unsigned, 4)
 	real f64 = 24.0
 	real += 6.0
 	real -= 2.0
@@ -9239,25 +9250,28 @@ compound_assignment_preserves_checked_numeric_operations :: proc(t: ^testing.T)
 	testing.expect_value(t, operation_counts[.Add], 3)
 	testing.expect_value(t, operation_counts[.Sub], 3)
 	testing.expect_value(t, operation_counts[.Mul], 3)
-	testing.expect_value(t, operation_counts[.Div], 3)
+	testing.expect_value(t, operation_counts[.Div], 1)
 
 	add_count := 0
 	sub_count := 0
 	mul_count := 0
 	div_count := 0
+	div_trunc_count := 0
 	for instruction in ir_module.functions[0].instructions {
 		#partial switch instruction.op {
 		case .Add_Checked: add_count += 1
 		case .Sub_Checked: sub_count += 1
 		case .Mul_Checked: mul_count += 1
 		case .Div_Checked: div_count += 1
+		case .Div_Trunc_Checked: div_trunc_count += 1
 		case:
 		}
 	}
 	testing.expect_value(t, add_count, 3)
 	testing.expect_value(t, sub_count, 3)
 	testing.expect_value(t, mul_count, 3)
-	testing.expect_value(t, div_count, 3)
+	testing.expect_value(t, div_count, 1)
+	testing.expect_value(t, div_trunc_count, 2)
 }
 
 @(test)
@@ -9310,7 +9324,7 @@ binary_arithmetic_rejects_non_numeric_operands :: proc(t: ^testing.T) {
 	text := `main func() i32 {
 	a i32 = 1
 	b u32 = 2
-	_ = a / b
+	_ = a + b
 	return 0
 }
 `
@@ -9366,7 +9380,7 @@ checked_division_and_subtraction_emit_guarded_llvm :: proc(t: ^testing.T) {
 	a i32 = 10
 	b i32 = 3
 	c i32 = a - b
-	return c / b
+	return div_trunc(c, b)
 }
 `
 	source_file := source.Source{path="test.bro", text=text}
@@ -9392,6 +9406,313 @@ checked_division_and_subtraction_emit_guarded_llvm :: proc(t: ^testing.T) {
 	testing.expect(t, strings.contains(llvm_text, "divovf_trap"))
 }
 
+@(test)
+integer_slash_is_rejected_and_float_slash_remains_available :: proc(t: ^testing.T) {
+	Case :: struct {text, want: string}
+	invalid := []Case{
+		{text=`main func() void {
+	a i32 = 4
+	b i32 = 2
+	_ = a / b
+}`, want="integer '/' is not allowed"},
+		{text=`main func() void {
+	a u32 = 4
+	b u32 = 2
+	_ = a / b
+}`, want="integer '/' is not allowed"},
+		{text=`main func() void {
+	_ = 4 / 2
+}`, want="integer '/' is not allowed"},
+		{text=`main func() void {
+	values [4 / 2]u8 = undefined
+	_ = &values
+}`, want="integer '/' is not allowed"},
+		{text=`half func($value i32) i32 { return value / 2 }
+main func() void { _ = $half(4) }`, want="integer '/' is not allowed"},
+		{text=`main func() void {
+	value i32 = 8
+	value /= 2
+}`, want="assign through an explicit division builtin"},
+	}
+	for test_case in invalid {
+		source_file := source.Source{path="test.bro", text=test_case.text}
+		diagnostics := source.init_diagnostics(&source_file)
+		symbols := symbol.init_table()
+		stream := lexer.lex(&source_file, &diagnostics, &symbols)
+		ast_module := parser.parse(&stream, &source_file, &diagnostics)
+		hir_module := checker.check(&ast_module, &diagnostics, &symbols)
+		found := false
+		for diagnostic in diagnostics.items {
+			found = found || strings.contains(diagnostic.message, test_case.want)
+		}
+		testing.expect(t, found)
+		hir.destroy_module(&hir_module)
+		ast.destroy_module(&ast_module)
+		delete(stream.items)
+		symbol.destroy_table(&symbols)
+		source.destroy_diagnostics(&diagnostics)
+	}
+
+	text := `main func() void {
+	value f32 = 5.0 / 2.0
+	value /= 2.0
+	_ = value
+}
+`
+	source_file := source.Source{path="test.bro", text=text}
+	diagnostics := source.init_diagnostics(&source_file)
+	defer source.destroy_diagnostics(&diagnostics)
+	symbols := symbol.init_table()
+	defer symbol.destroy_table(&symbols)
+	stream := lexer.lex(&source_file, &diagnostics, &symbols)
+	defer delete(stream.items)
+	ast_module := parser.parse(&stream, &source_file, &diagnostics)
+	defer ast.destroy_module(&ast_module)
+	hir_module := checker.check(&ast_module, &diagnostics, &symbols)
+	defer hir.destroy_module(&hir_module)
+	testing.expect_value(t, len(diagnostics.items), 0)
+}
+
+@(test)
+division_builtins_diagnose_arity_operands_and_comptime_failures :: proc(t: ^testing.T) {
+	text := `bad_arity :: div_floor(1)
+bad_bool :: rem(true, false)
+bad_family :: mod(i32(5), f32(3))
+zero_trunc :: div_trunc(1, 0)
+zero_floor :: div_floor(1.0, 0.0)
+zero_exact :: div_exact(1, 0)
+zero_ceil :: div_ceil(1.0, 0.0)
+zero_rem :: rem(1, 0)
+zero_mod :: mod(1.0, 0.0)
+inexact :: div_exact(5, 3)
+overflow_trunc :: div_trunc(min_value(i32), -1)
+overflow_floor :: div_floor(min_value(i32), -1)
+overflow_exact :: div_exact(min_value(i32), -1)
+overflow_ceil :: div_ceil(min_value(i32), -1)
+main func() void {}
+`
+	source_file := source.Source{path="test.bro", text=text}
+	diagnostics := source.init_diagnostics(&source_file)
+	defer source.destroy_diagnostics(&diagnostics)
+	symbols := symbol.init_table()
+	defer symbol.destroy_table(&symbols)
+	stream := lexer.lex(&source_file, &diagnostics, &symbols)
+	defer delete(stream.items)
+	ast_module := parser.parse(&stream, &source_file, &diagnostics)
+	defer ast.destroy_module(&ast_module)
+	hir_module := checker.check(&ast_module, &diagnostics, &symbols)
+	defer hir.destroy_module(&hir_module)
+	zero_count := 0
+	found_arity, found_operands, found_exact := false, false, false
+	overflow_count := 0
+	for diagnostic in diagnostics.items {
+		found_arity = found_arity || strings.contains(diagnostic.message, "expects 2 arguments")
+		found_operands = found_operands || strings.contains(diagnostic.message, "compatible numeric operands")
+		found_exact = found_exact || strings.contains(diagnostic.message, "exact division has a remainder")
+		if strings.contains(diagnostic.message, "signed integer division overflow") {
+			overflow_count += 1
+		}
+		if strings.contains(diagnostic.message, "division builtin denominator is zero") {
+			zero_count += 1
+		}
+	}
+	testing.expect(t, found_arity)
+	testing.expect(t, found_operands)
+	testing.expect(t, found_exact)
+	testing.expect_value(t, overflow_count, 4)
+	testing.expect_value(t, zero_count, 6)
+}
+
+@(test)
+division_family_compiles_and_runs_for_integer_and_float_scalars :: proc(t: ^testing.T) {
+	directory := "/tmp/brolang-test-division-family"
+	main_path := "/tmp/brolang-test-division-family/main.bro"
+	output := "/tmp/brolang-test-division-family-output"
+	text := `COUNT :: div_exact(8, 2)
+items [div_ceil(10, 3)]u8 :: [0, 0, 0, 0]
+OPEN :: 5
+open_ceil i32 :: div_ceil(OPEN, 3)
+
+check_i32 func(a, b, qt, qf, qc, r, m i32) bool {
+	return div_trunc(a, b) == qt and div_floor(a, b) == qf and
+	       div_ceil(a, b) == qc and rem(a, b) == r and mod(a, b) == m
+}
+
+check_f32 func(a, b, qt, qf, qc, r, m f32) bool {
+	return div_trunc(a, b) == qt and div_floor(a, b) == qf and
+	       div_ceil(a, b) == qc and rem(a, b) == r and mod(a, b) == m
+}
+
+check_f64 func(a, b, qt, qf, qc, r, m f64) bool {
+	return div_trunc(a, b) == qt and div_floor(a, b) == qf and
+	       div_ceil(a, b) == qc and rem(a, b) == r and mod(a, b) == m
+}
+
+edge_rem func(a, b i32) i32 { return rem(a, b) }
+edge_mod func(a, b i32) i32 { return mod(a, b) }
+
+main func() i32 {
+	if COUNT != 4 or items.len != 4 or open_ceil != 2 { return 1 }
+	if !check_i32(5, 3, 1, 1, 2, 2, 2) { return 2 }
+	if !check_i32(5, -3, -1, -2, -1, 2, -1) { return 3 }
+	if !check_i32(-5, 3, -1, -2, -1, -2, 1) { return 4 }
+	if !check_i32(-5, -3, 1, 1, 2, -2, -2) { return 5 }
+	if div_trunc(u32(5), u32(3)) != 1 or div_floor(u32(5), u32(3)) != 1 or
+	   div_ceil(u32(5), u32(3)) != 2 or rem(u32(5), u32(3)) != 2 or mod(u32(5), u32(3)) != 2 { return 6 }
+	if div_exact(i32(6), i32(3)) != 2 or div_exact(u32(6), u32(3)) != 2 { return 7 }
+	if !check_f32(f32(5.0), f32(3.0), f32(1.0), f32(1.0), f32(2.0), f32(2.0), f32(2.0)) or
+	   !check_f32(f32(5.0), f32(-3.0), f32(-1.0), f32(-2.0), f32(-1.0), f32(2.0), f32(-1.0)) or
+	   !check_f32(f32(-5.0), f32(3.0), f32(-1.0), f32(-2.0), f32(-1.0), f32(-2.0), f32(1.0)) or
+	   !check_f32(f32(-5.0), f32(-3.0), f32(1.0), f32(1.0), f32(2.0), f32(-2.0), f32(-2.0)) { return 8 }
+	if !check_f64(5.0, 3.0, 1.0, 1.0, 2.0, 2.0, 2.0) or
+	   !check_f64(5.0, -3.0, -1.0, -2.0, -1.0, 2.0, -1.0) or
+	   !check_f64(-5.0, 3.0, -1.0, -2.0, -1.0, -2.0, 1.0) or
+	   !check_f64(-5.0, -3.0, 1.0, 1.0, 2.0, -2.0, -2.0) { return 9 }
+	if div_exact(f32(6.0), f32(3.0)) != 2.0 or div_exact(f64(6.0), f64(3.0)) != 2.0 { return 10 }
+	if edge_rem(-2147483648, -1) != 0 or edge_mod(-2147483648, -1) != 0 { return 11 }
+	return 0
+}
+`
+	_ = os2.remove_all(directory)
+	defer _ = os2.remove_all(directory)
+	defer _ = os.remove(output)
+	testing.expect(t, os.make_directory(directory) == nil)
+	testing.expect(t, os.write_entire_file(main_path, transmute([]byte)text))
+	status := compiler_core.compile_package(directory, output)
+	testing.expect_value(t, status, 0)
+	state := run_executable(output)
+	testing.expect_value(t, state.exit_code, 0)
+}
+
+@(test)
+division_builtins_trap_for_runtime_zero_overflow_and_inexact_results :: proc(t: ^testing.T) {
+	Case :: struct {
+		name:      string,
+		type_name: string,
+		left:      string,
+		right:     string,
+	}
+	cases := [?]Case{
+		{name="div_trunc", type_name="i32", left="1", right="0"},
+		{name="div_floor", type_name="f32", left="f32(1.0)", right="f32(0.0)"},
+		{name="div_exact", type_name="f64", left="1.0", right="0.0"},
+		{name="div_ceil", type_name="i32", left="1", right="0"},
+		{name="rem", type_name="f32", left="f32(1.0)", right="f32(0.0)"},
+		{name="mod", type_name="f64", left="1.0", right="0.0"},
+		{name="div_exact", type_name="i32", left="5", right="3"},
+		{name="div_trunc", type_name="i32", left="-2147483648", right="-1"},
+		{name="div_floor", type_name="i32", left="-2147483648", right="-1"},
+		{name="div_exact", type_name="i32", left="-2147483648", right="-1"},
+		{name="div_ceil", type_name="i32", left="-2147483648", right="-1"},
+	}
+	for test_case, index in cases {
+		directory := fmt.aprintf("/tmp/brolang-test-division-trap-%d", index)
+		main_path := fmt.aprintf("%s/main.bro", directory)
+		output := fmt.aprintf("/tmp/brolang-test-division-trap-output-%d", index)
+		text := fmt.aprintf(
+			"invoke func(a, b %s) %s {{ return %s(a, b) }}\nmain func() void {{ _ = invoke(%s, %s) }}\n",
+			test_case.type_name, test_case.type_name, test_case.name, test_case.left, test_case.right,
+		)
+		_ = os2.remove_all(directory)
+		_ = os.remove(output)
+		testing.expect(t, os.make_directory(directory) == nil)
+		testing.expect(t, os.write_entire_file(main_path, transmute([]byte)text))
+		status := compiler_core.compile_package(directory, output)
+		testing.expect_value(t, status, 0)
+		state := run_executable(output)
+		testing.expect(t, !state.success)
+		_ = os.remove(output)
+		_ = os2.remove_all(directory)
+		delete(text)
+		delete(output)
+		delete(main_path)
+		delete(directory)
+	}
+}
+
+@(test)
+qualified_division_builtin_names_resolve_as_package_functions :: proc(t: ^testing.T) {
+	directory := "/tmp/brolang-test-qualified-division"
+	math_directory := "/tmp/brolang-test-qualified-division/math"
+	app_directory := "/tmp/brolang-test-qualified-division/app"
+	math_path := "/tmp/brolang-test-qualified-division/math/math.bro"
+	main_path := "/tmp/brolang-test-qualified-division/app/main.bro"
+	output := "/tmp/brolang-test-qualified-division-output"
+	math_text := `div_floor func(a, b i32) i32 { return a + b }
+`
+	main_text := `math :: import "../math"
+main func() i32 { return math.div_floor(20, 22) }
+`
+	_ = os2.remove_all(directory)
+	defer _ = os2.remove_all(directory)
+	defer _ = os.remove(output)
+	testing.expect(t, os.make_directory(directory) == nil)
+	testing.expect(t, os.make_directory(math_directory) == nil)
+	testing.expect(t, os.make_directory(app_directory) == nil)
+	testing.expect(t, os.write_entire_file(math_path, transmute([]byte)math_text))
+	testing.expect(t, os.write_entire_file(main_path, transmute([]byte)main_text))
+	status := compiler_core.compile_package(app_directory, output)
+	testing.expect_value(t, status, 0)
+	state := run_executable(output)
+	testing.expect_value(t, state.exit_code, 42)
+}
+
+@(test)
+division_builtins_emit_guards_rounding_and_single_integer_divisions :: proc(t: ^testing.T) {
+	text := `floor_i32 func(a, b i32) i32 { return div_floor(a, b) }
+ceil_i32 func(a, b i32) i32 { return div_ceil(a, b) }
+exact_i32 func(a, b i32) i32 { return div_exact(a, b) }
+floor_u32 func(a, b u32) u32 { return div_floor(a, b) }
+rem_i16 func(a, b i16) i16 { return rem(a, b) }
+mod_i16 func(a, b i16) i16 { return mod(a, b) }
+floor_f32 func(a, b f32) f32 { return div_floor(a, b) }
+ceil_f64 func(a, b f64) f64 { return div_ceil(a, b) }
+exact_f32 func(a, b f32) f32 { return div_exact(a, b) }
+rem_f64 func(a, b f64) f64 { return rem(a, b) }
+mod_f32 func(a, b f32) f32 { return mod(a, b) }
+main func() void {
+	_ = floor_i32(5, 3)
+	_ = ceil_i32(5, 3)
+	_ = exact_i32(6, 3)
+	_ = floor_u32(5, 3)
+	_ = rem_i16(5, 3)
+	_ = mod_i16(5, 3)
+	_ = floor_f32(f32(5.0), f32(3.0))
+	_ = ceil_f64(5.0, 3.0)
+	_ = exact_f32(f32(6.0), f32(3.0))
+	_ = rem_f64(5.0, 3.0)
+	_ = mod_f32(f32(5.0), f32(3.0))
+}
+`
+	source_file := source.Source{path="test.bro", text=text}
+	diagnostics := source.init_diagnostics(&source_file)
+	defer source.destroy_diagnostics(&diagnostics)
+	symbols := symbol.init_table()
+	defer symbol.destroy_table(&symbols)
+	stream := lexer.lex(&source_file, &diagnostics, &symbols)
+	defer delete(stream.items)
+	ast_module := parser.parse(&stream, &source_file, &diagnostics)
+	defer ast.destroy_module(&ast_module)
+	hir_module := checker.check(&ast_module, &diagnostics, &symbols)
+	defer hir.destroy_module(&hir_module)
+	ir_module := lower.lower(&hir_module)
+	defer ir.destroy_module(&ir_module)
+	llvm_text := llvm.emit(&ir_module, &diagnostics, &symbols)
+	defer delete(llvm_text)
+	testing.expect_value(t, len(diagnostics.items), 0)
+	testing.expect_value(t, strings.count(llvm_text, "sdiv i32"), 3)
+	testing.expect(t, !strings.contains(llvm_text, "srem i32"))
+	testing.expect(t, strings.contains(llvm_text, "udiv i32"))
+	testing.expect(t, strings.contains(llvm_text, "srem i16"))
+	testing.expect(t, strings.contains(llvm_text, "remspecial"))
+	testing.expect(t, strings.contains(llvm_text, "divzero_trap"))
+	testing.expect(t, strings.contains(llvm_text, "divovf_trap"))
+	testing.expect(t, strings.contains(llvm_text, "call float @llvm.floor.f32"))
+	testing.expect(t, strings.contains(llvm_text, "call double @llvm.ceil.f64"))
+	testing.expect(t, strings.contains(llvm_text, "call float @llvm.trunc.f32"))
+	testing.expect(t, strings.contains(llvm_text, "frem double"))
+}
+
 @(test)
 distinct_types_preserve_nominal_identity_and_backing_representation :: proc(t: ^testing.T) {
 	text := `Point :: struct {
diff --git a/examples/programs/compound_assignment/main.bro b/examples/programs/compound_assignment/main.bro
index 1dc442a..e4e4bb9 100644
--- a/examples/programs/compound_assignment/main.bro
+++ b/examples/programs/compound_assignment/main.bro
@@ -1,5 +1,4 @@
-# Milestone 6: compound assignment (`+=`, `-=`, `*=`, `/=`) and the binary
-# arithmetic operators `-`, `*`, `/` with multiplicative precedence.
+# Compound assignment (`+=`, `-=`, `*=`, `/=`) and explicit integer division.
 
 check_float func() i32 {
 	x f64 = 10.0
@@ -15,7 +14,7 @@ check_float func() i32 {
 
 check_unsigned func() i32 {
 	n u32 = 100
-	n /= 7 # 14 (truncating integer division)
+	n = div_trunc(n, 7) # 14
 	n -= 4 # 10
 	if n == 10 {
 		return 1
@@ -28,7 +27,7 @@ main func() i32 {
 	total += 10 # 10
 	total -= 3  # 7
 	total *= 4  # 28
-	total /= 2  # 14
+	total = div_trunc(total, 2) # 14
 
 	# binary operators honour precedence: 14 + (2 * 3) - 4 == 16
 	total = total + 2 * 3 - 4
diff --git a/std/arraylist/arraylist.bro b/std/arraylist/arraylist.bro
index 9ba3513..95cc061 100644
--- a/std/arraylist/arraylist.bro
+++ b/std/arraylist/arraylist.bro
@@ -30,7 +30,7 @@ reserve func($T type, list @mut ArrayList(T), minimum_capacity usize) void ! mem
 
 	new_capacity usize = 8
 	if list.capacity >= 8 {
-		half usize :: list.capacity / 2
+		half usize :: div_trunc(list.capacity, 2)
 		if list.capacity > max_value(usize) - half {
 			new_capacity = minimum_capacity
 		} else {
diff --git a/std/mem/mem.bro b/std/mem/mem.bro
index 2f3ad18..49450ed 100644
--- a/std/mem/mem.bro
+++ b/std/mem/mem.bro
@@ -61,7 +61,7 @@ alloc func($T type, allocator Allocator, count usize) []mut T ! AllocError {
 	if element_size == 0 {
 		return _empty_slice(T, count)
 	}
-	if count > max_value(usize) / element_size {
+	if count > div_trunc(max_value(usize), element_size) {
 		return .out_of_memory
 	}
 
@@ -86,7 +86,7 @@ realloc func($T type, allocator Allocator, memory []mut T, new_count usize) []mu
 	if element_size == 0 {
 		return _empty_slice(T, new_count)
 	}
-	if new_count > max_value(usize) / element_size {
+	if new_count > div_trunc(max_value(usize), element_size) {
 		return .out_of_memory
 	}
 
@@ -125,7 +125,7 @@ _power_of_two func(value usize) bool {
 
 	current usize = value
 	while current > 1 {
-		half usize = current / 2
+		half usize = div_trunc(current, 2)
 		if half * 2 != current {
 			return false
 		}