How to Use the Modulo Calculator

Select an operation type: basic modulo, modular arithmetic, congruence check, or practical applications. Enter your numbers and click Calculate.

Calculator Modes

Basic Modulo

Calculate the remainder when dividing one number by another:

a mod n: The remainder when a is divided by n
Quotient: How many times n fits into a
Division Expression: a = q × n + r

Modular Arithmetic

Advanced modular operations used in cryptography and number theory:

Modular Exponentiation (b^e mod m): Calculate large powers efficiently
Modular Inverse (a⁻¹ mod m): Find x where a × x ≡ 1 (mod m)

Congruence Check

Two numbers a and b are congruent modulo m (written a ≡ b (mod m)) if they have the same remainder when divided by m. Equivalently, (a - b) is divisible by m.

Applications

Practical uses of modulo arithmetic:

Luhn Algorithm: Validate check digits for credit cards, ISBNs, IMEIs
Day of Week: Calculate the weekday for any date using Zeller's congruence

What is Modulo?

The modulo operation (mod) returns the remainder after division. For example, 17 mod 5 = 2 because 17 = 3 × 5 + 2. It's fundamental in computer science and mathematics.

Common Applications

Cryptography (RSA, Diffie-Hellman)
Hash functions and data structures
Check digit validation (credit cards, barcodes)
Circular buffers and wrap-around logic
Calendar calculations
Random number generation

Key Properties

(a + b) mod n = ((a mod n) + (b mod n)) mod n
(a × b) mod n = ((a mod n) × (b mod n)) mod n
a ≡ b (mod n) means n divides (a - b)
Modular inverse exists only when gcd(a, n) = 1

Modulo Calculator