Test whether modulo is associative by comparing (a mod m) mod n vs a mod (m mod n). Side-by-side steps, visual bars, counter-example scanner, and coincidence detector.
The **Is Modulo Associative Calculator** investigates one of the most commonly misunderstood properties of the mod operation: **associativity**. In algebra, an operation ⊕ is associative if (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c) for all values. Addition and multiplication satisfy this property, but modulo does **not** — and this calculator proves it interactively.
Enter any triple (a, m, n) and the tool evaluates both groupings side by side: **(a mod m) mod n** on the left and **a mod (m mod n)** on the right. You'll see each intermediate step, the final values, and a color-coded visual comparison. Because the two sides usually differ, the calculator scans a range of triples to collect concrete counter-examples and, for completeness, identifies the occasional triples where the results coincidentally match.
Understanding why modulo is non-associative matters for compiler design (refactoring chained % expressions), cryptographic proofs (you cannot rearrange mod operations freely), and algorithm correctness (nested modular reductions must follow the proper order). This tool makes that lesson concrete with numbers, not just abstract algebra.
Many students assume modulo behaves like addition or multiplication and is safe to rearrange. This calculator disproves that assumption instantly by showing differing results for the two groupings and providing a table of counter-examples. It is an effective teaching aid for discrete-math or programming courses where modular-arithmetic properties are introduced.
Developers benefit too: a common bug is refactoring `(a % m) % n` into `a % (m % n)` — or vice versa — under the false assumption they are equivalent. Seeing many counter-examples at once makes the non-associativity memorable and helps prevent such mistakes in production code.
Left grouping: (a mod m) mod n. Right grouping: a mod (m mod n). If m mod n = 0 the right side is undefined. The two groupings are generally not equal.
Result: (20 mod 7) mod 3 = 0, but 20 mod (7 mod 3) = 0 — coincidence; try a = 10, m = 6, n = 4 for a clear difference.
For a = 10, m = 6, n = 4: left = (10 mod 6) mod 4 = 4 mod 4 = 0; right = 10 mod (6 mod 4) = 10 mod 2 = 0 — still matches by luck. For a = 20, m = 7, n = 3: left = 6 mod 3 = 0; right = 20 mod 1 = 0. The scanner finds many triples that differ.
Associativity requires the result to be independent of grouping for **all** inputs. Modulo fails because the two groupings reduce intermediate values in different orders. The left side first reduces a into the range [0, m), then further reduces into [0, n). The right side first reduces the modulus from m to m mod n, then reduces a by that smaller modulus. These two pipelines produce different residues unless the numbers align by coincidence.
In languages like JavaScript, C, Python, and Java, the `%` (or `mod`) operator is left-associative by default, so `a % m % n` parses as `(a % m) % n`. Introducing parentheses to form `a % (m % n)` changes behavior. Compiler optimizations that rearrange arithmetic must treat `%` differently from `+` and `*`, precisely because associativity does not hold.
Subtraction and division are the most familiar non-associative operations. Exponentiation is another: (2^3)^2 = 64 ≠ 2^(3^2) = 512. Modulo therefore belongs to a large family of useful but non-associative operations where parenthesization matters.
No. In general, (a mod m) mod n ≠ a mod (m mod n). Specific triples may coincidentally match, but the property does not hold universally.
Try a = 13, m = 6, n = 4: (13 mod 6) mod 4 = 1, while 13 mod (6 mod 4) = 13 mod 2 = 1. Because that still matches, it is a good reminder that the scanner matters. A different example such as a = 11, m = 7, n = 3 gives (11 mod 7) mod 3 = 1 and 11 mod (7 mod 3) = 11 mod 1 = 0.
If m mod n = 0, then a mod 0 is division by zero, so the right grouping has no value. That means the regrouped expression can fail even when the left-associated version is perfectly well-defined.
Yes. Refactoring chained % expressions by moving parentheses can silently introduce bugs. That is why modulo expressions should be treated more like subtraction or division than like addition.
No. a mod m ≠ m mod a in general. Modulo lacks both commutativity and associativity, so both the order and the grouping matter.
Addition, multiplication, logical AND, logical OR, string concatenation, and function composition are all associative. Those are the kinds of operations where regrouping does not change the final value.