Modular Arithmetic (Clock) Calculator

This modular arithmetic calculator works out what a number becomes when you wrap it around a clock of any size, the everyday idea behind telling the time past twelve or counting days of the week. You enter a number (n) and a modulus (m), and the calculator returns the least non-negative residue, the value left over once every full cycle of m has been removed, shown as a clear n mod m equals result statement. Alongside that main result you get the number of full cycles the value completes and the modulus you set, so you can see exactly how the answer was reached rather than just the final figure. Negative numbers are handled properly too, so entering minus 3 with a modulus of 12 wraps around to 9, rather than returning an ordinary negative remainder. Try the defaults of 15 and 12 to see 15 mod 12 equals 3, because 15 is one full turn of the clock plus 3 left over. Modular arithmetic like this sits behind clocks and calendars, check digits on ISBNs and bank cards, hashing algorithms and cryptography, so it is useful for students checking homework, programmers debugging code, and anyone wanting a fast, reliable answer without working through the division by hand. Change either the number or the modulus and the result updates straight away.

Number n

Modulus m

15 mod 12 = 3

Full cycles1

Modulus12

The formula

n mod m is the remainder when n is divided by m, taken as the least non-negative value from 0 to m minus 1. Negative numbers wrap around: the calculator returns ((n mod m) + m) mod m.

Worked example

15 mod 12 = 3, because 15 is one full turn of 12 plus 3. And minus 3 mod 12 = 9, wrapping back round the clock. Enter 15 and 12 to confirm.

Frequently asked questions

What is modular arithmetic?

Arithmetic that wraps around after reaching a modulus m, like hours on a clock. The result is the remainder from 0 to m minus 1.

How are negatives handled?

They wrap to a positive residue, so minus 1 mod 12 is 11.

Where is it used?

Clocks and calendars, ISBN and card check digits, hashing, and cryptography.

Who this calculator is for

This calculator is for students, programmers and anyone needing a quick, reliable result.

What this calculator assumes

You enter valid input.
The standard algorithm is applied.
Results are rounded for display.

Formula and sources

Related calculators

Scientific Calculator: general scientific maths.
Percentage Calculator: percentages and changes.
Prime Factorisation Calculator: factor a number into primes.

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