[modulus] 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.

Modulus operator gives you the result in 'reduced residue system'. For example for mod 5 there are 5 integers counted: 0,1,2,3,4. In fact 19=12=5=-2=-9 (mod 7). The main difference that the answer is given by programming languages by 'reduced residue system'.

Step 1 : 5/7 = 0.71

Step 2 : Take the left side of the decimal , so we take 0 from 0.71 and multiply by 7 0*7 = 0;

Step # : 5-0 = 5 ; Therefore , 5%7 =5

It's just about the remainders. Let me show you how

10 % 5=0
9 % 5=4 (because the remainder of 9 when divided by 5 is 4)
8 % 5=3
7 % 5=2
6 % 5=1

5 % 5=0 (because it is fully divisible by 5)

Now we should remember one thing, mod means remainder so

4 % 5=4

but why 4? because 5 X 0 = 0 so 0 is the nearest multiple which is less than 4 hence 4-0=4

modulus is remainders system.

So 7 % 5 = 2.

5 % 7 = 5

3 % 7 = 3

2 % 7 = 2

1 % 7 = 1

When used inside a function to determine the array index. Is it safe programming ? That is a different question. I guess.

As you say, the % sign is used to take the modulus (division remainder).

In w3schools' JavaScript Arithmetic page we can read in the Remainder section what I think to be a great explanation

In arithmetic, the division of two integers produces a quotient and a remainder.

In mathematics, the result of a modulo operation is the remainder of an arithmetic division.

So, in your specific case, when you try to divide 7 bananas into a group of 5 bananas, you're able to create 1 group of 5 (quotient) and you'll be left with 2 bananas (remainder).

If 5 bananas into a group of 7, you won't be able to and so you're left with again the 5 bananas (remainder).

A novel way to find out the remainder is given below

Statement : Remainder is always constant

ex : 26 divided by 7 gives R : 5 

This can be found out easily by finding the number that completely divides 26 which is closer to the divisor and taking the difference of the both

13 is the next number after 7 that completely divides 26 because after 7 comes 8, 9, 10, 11, 12 where none of them divides 26 completely and give remainder 0.

So 13 is the closest number to 7 which divides to give remainder 0.

Now take the difference (13 ~ 7) = 5 which is the temainder.

Note: for this to work divisor should be reduced to its simplest form ex: if 14 is the divisor, 7 has to be chosen to find the closest number dividing the dividend.

lets put it in this way:
actually Modulus operator does the same division but it does not care about the answer , it DOES CARE ABOUT reminder for example if you divide 7 to 5 ,
so , lets me take you through a simple example:
think 5 is a block, then for example we going to have 3 blocks in 15 (WITH Nothing Left) , but when that loginc comes to this kinda numbers {1,3,5,7,9,11,...} , here is where the Modulus comes out , so take that logic that i said before and apply it for 7 , so the answer gonna be that we have 1 block of 5 in 7 => with 2 reminds in our hand! that is the modulus!!!
but you were asking about 5 % 7 , right ?
so take the logic that i said , how many 7 blocks do we have in 5 ???? 0
so the modulus returns 0...
that's it ...

As others have pointed out modulus is based on remainder system.

I think an easier way to think about modulus is what remains after a dividend (number to be divided) has been fully divided by a divisor. So if we think about 5%7, when you divide 5 by 7, 7 can go into 5 only 0 times and when you subtract 0 (7*0) from 5 (just like we learnt back in elementary school), then the remainder would be 5 ( the mod). See the illustration below.

   0
  ______
7) 5    
__-0____
   5

With the same logic, -5 mod 7 will be -5 ( only 0 7s can go in -5 and -5-0*7 = -5). With the same token -5 mod -7 will also be -5. A few more interesting cases:

5 mod (-3) = 2 i.e. 5 - (-3*-1)

(-5) mod (-3) = -2 i.e. -5 - (-3*1) = -5+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.

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.

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.

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;

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.