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Is Modulo Multiplication and Addition Associative, Distributive, and Commutative?

We investigate whether modulo is associative, distributive, and commutative — and what these terms mean.

Clear
Remainder2PHP and C convention — sign follows the dividend
Least non-negative residue2the usual mathematical convention
Quotient (truncated)3
Floor division3
Check5 × 3 + 2 = 17
Divides evenly?No

The method

a mod n is the remainder after dividing a by n

Two conventions for negatives

−17 mod 5 is either −2 or 3, depending on the convention. Most programming languages truncate towards zero and give −2; mathematics and Python floor instead and give 3. Both satisfy a = qn + r; they differ in which q they pick. Number theory always wants the non-negative answer.