How Does Modulus Divison Work

Question

I don\'t really understand how modulus division works. I was calculating 27 % 16 and wound up with 11 and I don\'t understand why.

I can\'t

Answer 1

The only important thing to understand is that modulus (denoted here by % like in C) is defined through the Euclidean division.

For any two (d, q) integers the following is always true:

d = ( d / q ) * q + ( d % q )

As you can see the value of d%q depends on the value of d/q. Generally for positive integers d/q is truncated toward zero, for instance 5/2 gives 2, hence:

5 = (5/2)*2 + (5%2) => 5 = 2*2 + (5%2) => 5%2 = 1

However for negative integers the situation is less clear and depends on the language and/or the standard. For instance -5/2 can return -2 (truncated toward zero as before) but can also returns -3 (with another language).

In the first case:

-5 = (-5/2)*2 + (-5%2) => -5 = -2*2 + (-5%2) => -5%2 = -1

but in the second one:

-5 = (-5/2)*2 + (-5%2) => -5 = -3*2 + (-5%2) => -5%2 = +1

As said before, just remember the invariant, which is the Euclidean division.

Further details:

• What is the behavior of integer division?
• Division and Modulus for Computer Scientists

Answer 2

27 % 16 = 11

You can interpret it this way:

16 goes 1 time into 27 before passing it.

16 * 2 = 32.

So you could say that 16 goes one time in 27 with a remainder of 11.

In fact,

16 + 11 = 27

An other exemple:

20 % 3 = 2

Well 3 goes 6 times into 20 before passing it.

3 * 6 = 18

To add-up to 20 we need 2 so the remainder of the modulus expression is 2.

Answer 3

The modulus operator takes a division statement and returns whatever is left over from that calculation, the "remaining" data, so to speak, such as 13 / 5 = 2. Which means, there is 3 left over, or remaining from that calculation. Why? because 2 * 5 = 10. Thus, 13 - 10 = 3.

The modulus operator does all that calculation for you, 13 % 5 = 3.

Answer 4

It's simple, Modulus operator(%) returns remainder after integer division. Let's take the example of your question. How 27 % 16 = 11? When you simply divide 27 by 16 i.e (27/16) then you get remainder as 11, and that is why your answer is 11.

Answer 5

I hope these simple steps will help:

20 % 3 = 2 

1. 20 / 3 = 6; do not include the .6667 – just ignore it
2. 3 * 6 = 18
3. 20 - 18 = 2, which is the remainder of the modulo

Answer 6

Modulus division is pretty simple. It uses the remainder instead of the quotient.

    1.0833... <-- Quotient
   __
12|13
   12
    1 <-- Remainder
    1.00 <-- Remainder can be used to find decimal values
     .96
     .040
     .036
     .0040 <-- remainder of 4 starts repeating here, so the quotient is 1.083333...

13/12 = 1R1, ergo 13%12 = 1.

---

It helps to think of modulus as a "cycle".

In other words, for the expression n % 12, the result will always be < 12.

That means the sequence for the set 0..100 for n % 12 is:

{0,1,2,3,4,5,6,7,8,9,10,11,0,1,2,3,4,5,6,7,8,9,10,11,0,[...],4}

In that light, the modulus, as well as its uses, becomes much clearer.