Modulo Calculator

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

Worked Example: Modular Exponentiation

Suppose you need 7¹⁰ mod 13. Computing 7¹⁰ = 282,475,249 first would work here, but for cryptographic sizes (hundreds of digits) it is impossible. Instead, reduce at every step: 7² = 49 ≡ 10 (mod 13), 7⁴ ≡ 10² = 100 ≡ 9, 7⁸ ≡ 9² = 81 ≡ 3, and finally 7¹⁰ = 7⁸ × 7² ≡ 3 × 10 = 30 ≡ 4 (mod 13). This square-and-multiply method is exactly what the calculator uses, so it handles very large exponents instantly.