# Modulo Calculator

## Calculator Use

Calculate *a mod b* which, for positive numbers, is the remainder of *a* divided by *b* in a division problem. The modulo operation finds the remainder, so if you were dividing *a* by *b* and there was a remainder of *n*, you would say *a mod b = n*.

## How to Do a Modulo Calculation

The modulo operation finds the remainder of *a* divided by *b*. To do this by hand just divide two numbers and note the remainder. If you needed to find 27 mod 6, divide 27 by 6.

- 27 mod 6 = ?
- 27 ÷ 6 = 4 with a remainder of 3
- 27 mod 6 = 3

## Example Modulo Calculation

You need to write a piece of software that tells a user whether a number they input is a multiple of 4. You can use the modulo calculation to accomplish this.

If a number is a multiple of 4, when you divide it by 4 the remainder will be 0. So you would create the logic to take an input and use the *mod 4* operation on it. If the result is 0 the number is a multiple of 4 otherwise the number is not a multiple of 4.

The logic for this part of your program would be:

- x is the number input by the user
- If x mod 4 = 0 then x is a multiple of 4
- Else x is not a multiple of 4

If you did not use the mod operator you would have to do the math in your code. For example you would have to calculate "is 496 a multiple of 4?". You would divide 496 by 4, so 496 / 4 = 124 with no remainder. In terms of mod, 496 mod 4 = 0, so yes, 496 is a multiple of 4.

Is 226 a multiple of 4? Divide 226 by 4, so 226 / 4 = 56 with a remainder of 2. 226 mod 4 = 2, so no, 226 is not a multiple of 4.

In some calculators and computer programming languages a % b is the same as a mod b is the same as a modulo b where % or mod are used as the modulo operators.

## Example: 1 mod 2

*1 mod 2* is a situation where the divisor, 2, is larger than the dividend, 1, so the remainder you get is equal to the dividend, 1.

For 1 divided by 2, 2 goes into 1 zero times with a remainder of 1. So *1 mod 2 = 1*.

Similarly, *5 mod 10 = 5* since 10 divides into 5 zero times with 5 left over as the remainder.

For positive numbers, whenever the divisor (modulus) is greater than the dividend, the remainder is the same as the dividend.

### Further Reading

Explore modular arithmetic and modulo operations further including *a mod b* for negative numbers.

Kahn Academy, What is Modular Arithmetic?

Better Explained, Fun with Modular Arithmetic

Wikipedia, Applications of Modular Arithmetic

Mathworld, Congruence

Kahn Academy, Congruence Modulo