# What is modular arithmetic?

## An Introduction to Modular Math

When we divide two integers we will have an equation that looks like the following:

is the dividend

is the divisor

is the quotient

is the remainder

is the divisor

is the quotient

is the remainder

Sometimes, we are only interested in what the

For these cases there is an operator called the modulo operator (abbreviated as mod).

**remainder**is when we divide by .For these cases there is an operator called the modulo operator (abbreviated as mod).

Using the same , , , and as above, we would have:

We would say this as

*modulo**is equal to*. Where is referred to as the**modulus**.For example:

## Visualize modulus with clocks

Observe what happens when we increment numbers by one and then divide them by 3.

The remainders start at 0 and increases by 1 each time, until the number reaches one less than the number we are dividing by. After that, the sequence

**repeats**.By noticing this, we can visualize the modulo operator by using circles.

We write 0 at the top of a circle and continuing clockwise writing integers 1, 2, ... up to one less than the modulus.

For example, a clock with the 12 replaced by a 0 would be the circle for a modulus of 12.

To find the result of we can follow these steps:

- Construct this clock for size
- Start at 0 and move around the clock steps
- Wherever we land is our solution.

(If the number is positive we step clockwise, if it's

**negative**we step**counter-clockwise**.)## Examples

With a modulus of 4 we make a clock with numbers 0, 1, 2, 3.

We start at 0 and go through 8 numbers in a clockwise sequence 1, 2, 3, 0, 1, 2, 3, 0.

We start at 0 and go through 8 numbers in a clockwise sequence 1, 2, 3, 0, 1, 2, 3, 0.

We ended up at

**0**so .With a modulus of 2 we make a clock with numbers 0, 1.

We start at 0 and go through 7 numbers in a clockwise sequence 1, 0, 1, 0, 1, 0, 1.

We start at 0 and go through 7 numbers in a clockwise sequence 1, 0, 1, 0, 1, 0, 1.

We ended up at

**1**so .With a modulus of 3 we make a clock with numbers 0, 1, 2.

We start at 0 and go through 5 numbers in

We start at 0 and go through 5 numbers in

**counter-clockwise**sequence (5 is**negative**) 2, 1, 0, 2, 1.We ended up at

**1**so .## Conclusion

If we have and we increase by a

**multiple of**, we will end up in the same spot, i.e.forany integer.

For example:

## Notes to the Reader

### mod in programming languages and calculators

Many programming languages, and calculators, have a mod operator, typically represented with the % symbol. If you calculate the result of a negative number, some languages will give you a negative result.

e.g.

e.g.

`-5 % 3 = -2.`

### Congruence Modulo

You may see an expression like:

$A \equiv B\ (\text{mod } C)$

This says that is

**congruent**to modulo . It is similar to the expressions we used here, but not quite the same.In the next article we will explain what it means and how it is related to the expressions above.