Modulo & Remainder
MathematicsCalculate integer division quotient, remainder, and mathematical modular congruence (a mod b). Essential for cryptography, clock arithmetic, and computer programming.
a = 3 × b + 2
About this calculator
Modular arithmetic (often called clock arithmetic) forms the mathematical backbone of modern public-key cryptography (RSA, Diffie-Hellman), hashing algorithms, and cyclic scheduling in software systems.
This tool evaluates both standard truncated programming remainders and true Euclidean mathematical modulo, which always produces a non-negative result even for negative dividends.
How It Works & Formula
Euclidean modulo computes r = a - b * floor(a / b), ensuring 0 ≤ r < |b|. Standard integer division computes quotient q = trunc(a / b) and remainder r_trunc = a % b.
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