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 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.

Two Steps Solution.

Some of the answers here are complicated for me to understand. I will try to add one more answer in an attempt to simplify the way how to look at this.

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Short Answer:

Example 1:

7 % 5 = 2

Each person should get one pizza slice.

Divide 7 slices on 5 people and every one of the 5 people will get one pizza slice and we will end up with 2 slices (remaining). 7 % 5 equals 2 is because 7 is larger than 5.

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Example 2:

5 % 7 = 5

Each person should get one pizza slice

It gives 5 because 5 is less than 7. So by definition, you cannot divide whole 5items on 7 people. So the division doesn't take place at all and you end up with the same amount you started with which is 5.

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Programmatic Answer:

The process is basically to ask two questions:

Example A: (7 % 5)

(Q.1) What number to multiply 5 in order to get 7?

Two Conditions: Multiplier starts from `0`. Output result should not exceed `7`. 

Let's try:

Multiplier is zero 0 so, 0 x 5 = 0

Still, we are short so we add one (+1) to multiplier.

1 so, 1 x 5 = 5

We did not get 7 yet, so we add one (+1).

2 so, 2 x 5 = 10

Now we exceeded 7. So 2 is not the correct multiplier. Let's go back one step (where we used 1) and hold in mind the result which is5. Number 5 is the key here.

(Q.2) How much do we need to add to the 5 (the number we just got from step 1) to get 7?

We deduct the two numbers: 7-5 = 2.

So the answer for: 7 % 5 is 2;

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Example B: (5 % 7)

1- What number we use to multiply 7 in order to get 5?

Two Conditions: Multiplier starts from `0`. Output result and should not exceed `5`. 

Let's try:

0 so, 0 x 7 = 0

We did not get 5 yet, let's try a higher number.

1 so, 1 x 7 = 7

Oh no, we exceeded 5, let's get back to the previous step where we used 0 and got the result 0.

2- How much we need to add to 0 (the number we just got from step 1) in order to reach the value of the number on the left 5?

It's clear that the number is 5. 5-0 = 5

   5 % 7 = 5

Hope that helps.