Modular Arithmetic Calculator: Clock Math and Cr...

What is Modular Arithmetic?

Modular arithmetic is "clock math" — numbers wrap around after reaching a modulus (m). 7 mod 3 = 1 (same as 3+3+1). It is the mathematical foundation of cyclic phenomena, cryptographic systems (RSA, AES, Diffie-Hellman), error-detecting codes, and computer hash functions.

🔐 Modular Arithmetic Calculator

Free calculator for instant results.

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📐 Formula

a ≡ b (mod m) means m | (a−b)

a mod m = remainder when a is divided by m. Range: 0 to m−1. For negative numbers: −1 mod 5 = 4 (not −1).

📝 Worked Example

RSA key example: p=61, q=53, n=3233
φ(n)=(p−1)(q−1)=3120
Public key e=17, Private key d: 17d ≡ 1 (mod 3120)
d = 2753 (modular inverse)

📝 How to Use

Enter a and mCompute a mod m for basic modular reduction.

Modular InverseFind x such that ax ≡ 1 (mod m) using extended Euclidean algorithm.

Euler's TotientCompute φ(n) — count of integers 1..n coprime with n.

CRT SolverChinese Remainder Theorem — solve simultaneous congruences.

❓ FAQ

What is the modular inverse and when does it exist?

The inverse of a (mod m) exists if and only if GCD(a,m)=1 (a and m are coprime). Found using the Extended Euclidean Algorithm.

How is modular arithmetic used in RSA encryption?

RSA relies on: (mᵉ)ᵈ ≡ m (mod n). The difficulty of factoring large n = p×q makes the private key (d) computationally infeasible to find without knowing p and q.

What is Modular Arithmetic?