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.