# Calculator Shortcut for Modular Arithmetic

## Modular arithmetic

If you think back to doing division with whole numbers, you may remember finding the whole number result and the remainder after division.### Modulus

The**modulus**[footnote]Sometimes, instead of seeing 17 mod 5 = 2, you’ll see 17 ≡ 2 (mod 5). The ≡ symbol means “congruent to” and means that 17 and 2 are equivalent, after you consider the modulus 5.[/footnote] is another name for the remainder after division. For example, 17 mod 5 = 2, since if we divide 17 by 5, we get 3 with remainder 2.

### Example 1

Compute the following:- 10 mod 3
- 15 mod 5
- 27 mod 5

#### Answers

- Since 10 divided by 3 is 3 with remainder 1, 10 mod 3 = 1
- Since 15 divided by 5 is 3 with no remainder, 15 mod 5 = 0
- 2
^{7}= 128. 128 divide by 5 is 25 with remainder 3, so 2^{7}mod 5 = 3

### Try it Now

Compute the following:- 23 mod 7
- 15 mod 7
- 2034 mod 7

### Modulus on a Standard Calculator

To calculate*a*mod

*n*on a standard calculator

- Divide
*a*by*n* - Subtract the whole part of the resulting quantity
- Multiply by
*n*to obtain the modulus

### Example 2

Calculate 31345 mod 419.#### Answer

[latex]31345\div{419}=74.8090692[/latex] | Now subtract 74 to get just the decimal remainder |

[latex]74.8090692-74=0.8090692[/latex] | Multiply this by 419 to get the modulus |

[latex]0.8090692\times{419}=339[/latex] | This tells us 0.8090692 was equivalent to [latex]\frac{339}{419}[/latex] |