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Modulo Calculator

DeveloperMath

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INPUT

Auto Process

OUTPUT

Client Side

Result

Truncated Division (a mod n)

Euclidean Modulo (always ≥ 0)

Step-by-Step Breakdown

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Guide

Modulo Calculator

Calculate the remainder of a division operation instantly. The Modulo Calculator supports six operations — simple mod, modular addition, subtraction, multiplication, exponentiation, and congruence checking — with full BigInt support for arbitrarily large numbers and both truncated and Euclidean negative number handling.

How to Use

Select an operation from the dropdown, enter your values, and the result appears instantly. For simple modulo, enter a and n (modulus). For binary operations, enter a, b, and n. Use the step-by-step breakdown to understand exactly how the result was computed.

Features

6 operations – simple mod, modular add, subtract, multiply, exponentiate, and congruence check
BigInt support – handles arbitrarily large integers beyond JavaScript’s standard number limits
Negative number handling – both truncated division (C/Java style) and Euclidean (always non-negative) modes
Step-by-step breakdown – shows each computation step for educational purposes
Real-time results – output updates instantly as you type

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 FAQ

What is the difference between truncated and Euclidean modulo?

Truncated modulo follows the C/Java convention where the result has the same sign as the dividend: −10 mod 3 = −1. Euclidean modulo always returns a non-negative result: −10 mod 3 = 2. The Euclidean definition is mathematically preferred because it satisfies the property that a ≡ r (mod n) where 0 ≤ r < n.
What is modular arithmetic used for in computer science?

Modular arithmetic is fundamental to cryptography (RSA, Diffie-Hellman key exchange), hash functions, cyclic data structures like ring buffers, calendar calculations, checksums, and pseudo-random number generation. It is also used in error detection codes like ISBN, IBAN, and Luhn validation.
What does it mean for two numbers to be congruent modulo n?

Two integers a and b are congruent modulo n (written a ≡ b mod n) if they have the same remainder when divided by n, or equivalently if their difference (a − b) is divisible by n. For example, 17 ≡ 5 (mod 12) because 17 − 5 = 12, which is divisible by 12.
How does modular exponentiation work?

Modular exponentiation computes (baseⁿ) mod m efficiently using the repeated squaring algorithm, which reduces an exponential number of multiplications to a logarithmic number. This is critical in public-key cryptography where exponents can have hundreds of digits.