# What is modularÂ arithmetic?

## An Introduction to Modular Math

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

$A$ is the dividend

$B$ is the divisor

$Q$ is the quotient

$R$ is the remainder

$B$ is the divisor

$Q$ is the quotient

$R$ 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 $A$ by $B$.For these cases there is an operator called the modulo operator (abbreviated as mod).

Using the same $A$, $B$, $Q$, and $R$ as above, we would have: $A \text{ mod } B = R$

We would say this as $A$

*modulo*$B$*is equalÂ to*$R$. Where $B$ 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 $A \text{ mod } B$ we can follow these steps:

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

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

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

### $8 \text{ mod } 4 = ?$

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 $8 \text{ mod } 4 = \bf{0}$.### $7 \text{ mod } 2 = ?$

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 $7 \text{ mod } 2 = \bf{1}$.**$-5 \text{ mod } 3 = ?$**

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 $-5 \text{ mod } 3 = \bf{1}$.## Conclusion

If we have $A \text{ mod } B$Â and we increase $A$ by a

**multiple of $\bf{B}$**, we will end up in the same spot, i.e.$A \text{ mod } B = (A + K \cdot B) \text{ mod } B$Â forany integer $\bf{K}$.

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.Â 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 $A$ is

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