How do I prove that 5 divides x^5 - x?

[ninjagod123]

How do I prove that 5 divides x^5 - x??

How do I prove that 5 divides x^5 - x??

 

[CRGreathouse]

Compute the fifth powers mod 5.

 

[CompuChip]

Let me give you a hint: checking it for x = 0, 1, 2, 3 and 4 suffices.

(In fact, as 4 = -1 (mod 5) and 3 = -2 (mod 5), you can see that only x = 0, 1, 2 will do).

I'll let you figure out why.