# Understanding The Modulus Operator %

(This explanation is only for positive numbers since it depends on the language otherwise)

Definition

The Modulus is the remainder of the euclidean division of one number by another. `%` is called the modulo operation.

For instance, `9` divided by `4` equals `2` but it remains `1`. Here, `9 / 4 = 2` and `9 % 4 = 1`.

In your example: 5 divided by 7 gives 0 but it remains 5 (`5 % 7 == 5`).

Calculation

The modulo operation can be calculated using this equation:

```a % b = a - floor(a / b) * b

```
• `floor(a / b)` represents the number of times you can divide `a` by `b`
• `floor(a / b) * b` is the amount that was successfully shared entirely
• The total (`a`) minus what was shared equals the remainder of the division

Applied to the last example, this gives:

```5 % 7 = 5 - floor(5 / 7) * 7 = 5

```

Modular Arithmetic

That said, your intuition was that it could be -2 and not 5. Actually, in modular arithmetic, `-2 = 5 (mod 7)` because it exists k in Z such that `7k - 2 = 5`.

You may not have learned modular arithmetic, but you have probably used angles and know that -90° is the same as 270° because it is modulo 360. It’s similar, it wraps! So take a circle, and say that it’s perimeter is 7. Then you read where is 5. And if you try with 10, it should be at 3 because `10 % 7` is 3.