# How do I prove that 5 divides x^5 – x? The joys of modular arithmetic

Posted by: Gary Ernest Davis on: February 11, 2010

“How do I prove that 5 divides $x^5 - x$??”

In this question, the variable “$x$” should be thought of as restricted to integer values – in fact positive integer values – since divisibility for fractions or real numbers in general is trivial.

So another way to ask the question is: if $x$ is a positive integer, how do we see that $x^5-x$ is always a multiple of 5?

First thought is to factor an $x$ in  $x^5-x$ to get $x(x^4-1)$.

The reason this is suggestive is because 5 is a prime number , so if 5 divides  $x(x^4-1)$ then either 5 divides $x$ or 5 divides $x^4-1$.

So now the question becomes, if 5 does not divide $x$, why does 5 divide $x^4-1$?

We can think of this in terms of arithmetic modulo 5, where we only keep remainders after division by 5.

In the language of modular arithmetic, the question is:

if $x\neq 0 (mod 5)$ is $x^4-1= 0 (mod 5)$?

$x\neq 0 (mod 5)$ means $x$ gives a remainder of 1, 2, 3 or 4 after division by 5 , so there are 4 cases to consider.

How about the other way around:

if $x^4-1\neq 0 (mod 5)$ is $x =0 (mod 5)$?

$x^4-1\neq 0 (mod 5)$ means $x^4 \neq 1 (mod 5)$

What are the $4^{th}$ powers of integers modulo 5?

 x x^4 x^4 (mod 5) 0 0 0 1 1 1 2 16 1 3 81 1 4 256 1

So this little table tells us that the only time $x^4 \neq 1 (mod 5)$

is when $x$ is itself a multiple of 5.

This simple little argument shows the power of working with remainders after whole number division – which is just modular arithmetic.

### Fermat’s little theorem

Piere de Fermat stated, and Euler and others proved, a useful little gem of number theory, known as Fermat’s little theorem.

It states that if $p$ is a prime number then for all integers $x$, $x^p = x(mod p)$.

Since 5 is a prime number, the result above is just a special case of Fermat’s little theorem.

This page has a number of different proofs of Fermat’s little theorem.

### 2 Responses to "How do I prove that 5 divides x^5 – x? The joys of modular arithmetic"

Are we allowed to use mathematical induction?

I don’t think anyone’s stopping you, are they?