How can I make 5 mod 5 = 5 instead of zero?
discrete mathematics - Why is $a^{5} \equiv a\pmod 5$ for any positive integer? - Mathematics Stack Exchange
Understanding The Modulus Operator % - Stack Overflow
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How much is 17 mod 3?
17 mod 3 equals 2 since dividing 17 by 3 gives a quotient of 5 and a remainder of 2. The remainder is the result of the modulus operation. In simpler terms, 17 mod 3 = 2.
What is modulo (mod) operation?
What is the difference between mod and remainder?
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var mod5 = function (xx) {
return (xx % 5) || 5
}
This function will return xx mod 5 if it is different of zero, otherwise it will return 5, more explained:
xx % 5 will result in any number between -4 and 4 -4 <= xx % 5 <= 4, then, an OR logical operator compares that result with the number five, the OR logical operator works as the following:
If the value in the left side of the operator is true, or truthy it will be returned independently of the value in the right side.
If the value in the right side is true, or truthy, AND the value in the left side is false, or falsy it will be returned.
Orherwise, in case none of the value is true or truthy, false will be returned.
In this case, xx % 5 will be returned only if xx is not multiple of 5, otherwise, 5 will be returned.
Hope that helps :)
function mod5(x) {
var result = x % 5;
if (result === 0) return 5;
return result;
}
Per your requirements, anything that is divisible by 5 will yield 5. Everything else is unchanged.
OK, without using Fermat's Little Theorem (a far more general and elegant result), here's another easy workaround.
Any integer 
can be exactly one of 
.
Take the fifth powers of each of those and see them reduce back to the original residue in each case.
One way to prove this is to prove it by induction. If 
show that 
.
Note that 
The general theorem people have mentioned in comments, Fermat's Little Theorem, states that if 
is prime and 
is any number: 
.
(This explanation is only for positive numbers since it depends on the language otherwise)
Definition
The modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the modulus of the operation. (source: wikipedia)
For instance, 9 divided by 4 equals 2 but it remains 1. Here, 9 / 4 = 2 and 9 % 4 = 1.
Image source: Wikimedia
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 divideabybfloor(a / b) * bis 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.
Short Answer:
Example 1:
7 % 5 = 2Each 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 = 5Each 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.