Modulo Calculator

Name: Modulo Calculator
Author: Jitendra Kumar

Calculate integer remainders, quotients, divisibility, congruence checks, and simple modular addition or multiplication.

Last Updated: May 2026

Remainder

Quotient

Divisibility

Not divisible

Status

Late in the cycle

Modulo Inputs

Enter integers. The calculator uses Euclidean modulo, so the remainder is always between 0 and the positive modulus.

Dividend (A)

Modulus (B)

Compare value

Used for congruence check.

Addend

Multiplier

Division Details

Step	Calculation	Result
Euclidean division	125 = 7 x 17 + 6	125 mod 7 = 6
Remainder bounds	0 <= 6 < 7	Valid
Lower multiple	17 x 7	119
Next multiple	119 + 7	126
Distance to next multiple	modulus - remainder	1

Congruence Check

Check	Calculation	Result
Compare remainder	20 mod 7	6
Congruence check	125 and 20 modulo 7	Congruent
Difference test	125 - 20	Divisible by modulus
Negative residue	Equivalent signed residue	-1

Modular Operations

Operation	Calculation	Result
Modular addition	(6 + 4) mod 7	3
Modular multiplication	(6 x 5) mod 7	2
Programming remainder	125 % 7	6

Modulo Convention Notice

This calculator uses Euclidean modulo with a nonnegative remainder. Some programming languages use a remainder operator that can return a negative value for negative dividends, so check your language’s convention when translating results into code.

Checked by Jitendra Kumar

Modulo Calculator is checked for formula labels, source links, and result limits.

Jitendra Kumar, Founder & Editorial Standards Lead. Updated May 2026. Scope: math calculators.

Sources & methodology · Review standards

How to Use the Modulo Calculator

Enter a dividend and a modulus. The calculator returns the Euclidean remainder, quotient, divisibility status, nearest multiples, and the division identity.

Add a compare value to test whether two numbers are congruent modulo the same modulus. Add an addend or multiplier to preview modular arithmetic operations.

Step 1: Enter the dividend
This is the number being divided.
Step 2: Enter the modulus
This is the positive cycle size or divisor. It cannot be zero.
Step 3: Review the remainder
The remainder is shown with the quotient and A = BQ + R identity.
Step 4: Check congruence or operations
Use the compare, addend, and multiplier fields for modular arithmetic context.

How This Modulo Calculator Works

Modulo arithmetic keeps the remainder after division. If a number can be written as A = BQ + R with 0 <= R < B, then A mod B equals R.

For negative dividends, this calculator adjusts the raw remainder into the standard nonnegative range. That is why -16 mod 26 becomes 10: -16 = 26 x -1 + 10.

Congruence checks use the same idea. If two values have the same remainder modulo B, their difference is divisible by B and they are congruent modulo B.

Modulo and Remainders Guide

Core Formulas

Concept	Formula	Use
Modulo	A mod B = R	R is the remainder after dividing A by B.
Euclidean division	A = BQ + R, where 0 <= R < B	Defines quotient Q and remainder R.
Quotient	Q = (A - R) / B	Integer number of whole modulus groups.
Divisibility	A mod B = 0	A is evenly divisible by B.
Congruence	A ≡ C (mod B) when B divides A - C	A and C have the same remainder.

Examples

Expression	Division identity	Remainder
13 mod 5	13 = 5 x 2 + 3	3
8 mod 4	8 = 4 x 2 + 0	0
-16 mod 26	-16 = 26 x -1 + 10	10
40 mod 12	40 = 12 x 3 + 4	4

Modulo Context

Modulo is useful whenever values wrap around a fixed cycle. Clocks, weekday schedules, repeating patterns, circular buffers, and hash tables all use this idea.

In number theory, modulo notation is also used for congruences. Saying two numbers are congruent modulo m means they land in the same remainder class after division by m.

Keep the research moving with LCM / GCF Calculator, Numbers Converter, Scientific Calculator, and Mean Calculator.

Frequently Asked Questions

Modulo gives the remainder after division. For example, 13 mod 5 equals 3 because 13 is 5 times 2 plus 3.

Use A = BQ + R, where A is the dividend, B is the modulus, Q is the quotient, and R is the remainder with 0 <= R < B.

Yes. This calculator uses Euclidean modulo, so the remainder stays nonnegative. For example, -16 mod 26 equals 10.

Some languages use a remainder operator whose sign follows the dividend. The calculator shows that programming-style remainder separately when it differs from Euclidean modulo.

Two integers are congruent modulo m when they have the same remainder after division by m, or equivalently when their difference is divisible by m.

Modulo is used for clock arithmetic, divisibility checks, cyclic patterns, hashing, calendars, number theory, cryptography, and programming.