Modulo Calculator

Modulo Arithmetic Guide

What is Modulo?

The modulo operation (written a mod n or a % n in programming) gives the remainder after dividing a by n. 17 mod 5 = 2, because 17 = 5×3 + 2. 20 mod 4 = 0 (20 is exactly divisible by 4). 7 mod 7 = 0. For any integer a and positive n: a mod n is always between 0 and n-1. The modulo operation is fundamentally different from division — it discards the quotient and keeps only the remainder.

Negative Numbers and Modulo

Different programming languages handle negative modulo differently. In Python: -7 mod 3 = 2 (mathematical/floored modulo — always non-negative). In C, Java, JavaScript: -7 % 3 = -1 (truncated remainder — takes sign of dividend). The mathematical definition (result always non-negative) is generally more useful. -7 mod 3 = 2 because -7 = 3×(-3) + 2. This calculator uses the mathematical (non-negative) convention.

Modular Arithmetic Applications

Clock arithmetic: 'It is 10am, what time will it be in 15 hours?' → (10 + 15) mod 24 = 1am. Days of the week: day 0 = Sunday, if today is Wednesday (day 3), what day is it in 100 days? → (3 + 100) mod 7 = 5 = Friday. Calendar calculations: any repeating cycle uses modulo. Cryptography: RSA encryption is based on modular exponentiation. Checksums: IBAN bank account validation, ISBN book number checks.

In Programming

The % operator in most programming languages gives the remainder (modulo). Checking if a number is even: n % 2 == 0. Checking if a number is divisible by k: n % k == 0. Wrapping around an array: index = (index + 1) % arrayLength. Converting between 12-hour and 24-hour time: hour12 = hour24 % 12 || 12. Generating a hash that fits in a table of size n: hash = value % n. Modulo is one of the most frequently used operations in programming.