➗Modulo Calculator

About this tool

Enter a and n to see the remainder under all three conventions and the quotient. The second section computes bᵉ mod m, and the third finds a⁻¹ mod m. All three accept integers of any size.

How modulo works

a mod n is the remainder after dividing a by n: 17 mod 5 = 2, because 17 = 3 × 5 + 2. It powers clock arithmetic (15:00 is 3 o'clock), days of the week, hash tables, check digits and cryptography. For negative numbers, definitions differ. Truncated division, used by the % operator in C, C++, Java and JavaScript, gives −7 % 3 = −1. Floored division (Python, Ruby) gives 2, taking the sign of n. The Euclidean definition preferred in maths always gives a remainder between 0 and |n| − 1, so here also 2. Modular exponentiation, bᵉ mod m, uses repeated squaring, so even 3^200 mod 13 (= 9) takes only a handful of steps; it's the core operation of RSA and Diffie–Hellman. The modular inverse a⁻¹ is the number that gives a × a⁻¹ ≡ 1 (mod m), found with the extended Euclidean algorithm. It exists only when a and m are coprime: 3⁻¹ mod 11 = 4, but 4 has no inverse mod 8. The GCD and LCM calculator and the prime number checker are related.

Frequently asked questions