Understanding The Modulus Operator %

I understand the Modulus operator in terms of the following expression:

7 % 5

This would return 2 due to the fact that 5 goes into 7 once and then gives the 2 that is left over, however my confusion comes when you reverse this statement to read:

5 % 7

This gives me the value of 5 which confuses me slightly. Although the whole of 7 doesn't go into 5, part of it does so why is there either no remainder or a remainder of positive or negative 2?

If it is calculating the value of 5 based on the fact that 7 doesn't go into 5 at all why is the remainder then not 7 instead of 5?

I feel like there is something I'm missing here in my understanding of 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 its 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.