Modulo Calculator (a mod n, Modular Inverse, Modular Power)

The modulo operation gives the remainder after division: 17 mod 5 = 2 because 17 = 3 × 5 + 2. For positive numbers everyone agrees, but with negatives, programming languages give different answers, which causes real bugs. This calculator shows all three conventions side by side. It also finds modular inverses and modular powers of integers with hundreds of digits, the arithmetic behind clock maths, checksums and RSA.

How to use this tool

1. Choose what to work out: a remainder, an inverse, a power or a congruence check.
2. Enter a and the modulus n as whole numbers; negative values are allowed for remainders.
3. For a power, enter the exponent b; for a congruence check, enter the second number b.
4. Read the result and the step table (Euclid steps or square-and-multiply bits).

Formula

a = q × n + r · a⁻¹: a × a⁻¹ ≡ 1 (mod n) · a^b mod n by repeated squaring

Truncated division rounds the quotient q toward zero, so r takes the sign of a (C, Java, JavaScript). Floored division rounds q down, so r takes the sign of n (Python, Excel MOD). The Euclidean remainder is always between 0 and |n| − 1, the usual choice in number theory. The inverse comes from the extended Euclidean algorithm and exists only when gcd(a, n) = 1. Powers use square-and-multiply, so even a 1,000-digit exponent needs only a few thousand steps.

Example

For −7 mod 3, JavaScript's -7 % 3 gives −1 (−7 = −2 × 3 − 1), while Python's -7 % 3 gives 2 (−7 = −3 × 3 + 2); the Euclidean answer is also 2. In RSA with e = 17 and φ = 3120, the inverse of 17 mod 3120 is 2753, since 17 × 2753 = 46,801 = 15 × 3120 + 1. And 4¹³ mod 497 = 445.