Modulo Calculator

About the Modulo Calculator

The modulo operation finds what is left over after dividing one whole number by another. 17 mod 5 is 2, because 17 = 5 x 3 + 2. For positive numbers the mathematical modulo and the % operator in programming agree, but for negative numbers they do not. In mathematics, and in Python, a mod n always has the sign of n, so -7 mod 3 is 2. In JavaScript, Java and C, the % operator is a remainder that has the sign of the first number, so -7 % 3 is -1. The calculator shows both together with the floor and truncated quotients that go with them, and each satisfies a = n x quotient + result. The modular inverse of a modulo n is the number x for which a x leaves 1 when divided by n. It exists only when a and n share no common factor, for example the inverse of 3 mod 11 is 4 because 3 x 4 = 12 and 12 mod 11 = 1. It is found with the extended Euclidean algorithm, and the steps are shown. The power tab computes a to the power b mod n by repeated squaring, so even exponents with hundreds of digits are fast; for example 2 to the 10 mod 1000 is 24. A negative exponent uses the inverse. Two numbers are congruent modulo n when n divides their difference. These operations are the building blocks of clock arithmetic, hashing, checksums, calendars and public key cryptography.