SOLUTION: Prove that x to the power of five minus x is divisible by five

SOLUTION: Prove that x to the power of five minus x is divisible by five

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Question 847612: Prove that x to the power of five minus x is divisible by five
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
Assuming x is an integer, a somewhat brute-force but valid solution is to check each of the residues 0 - 4:

0^5 - 0 = 0 ≡ 0 (mod 5)
1^5 - 1 = 0 ≡ 0
2^5 - 2 = 30 ≡ 0
3^5 - 3 = 240 ≡ 0
4^5 - 4 = 1020 ≡ 0

Since we have checked each of the residues, x^5 - x is divisible by 5 for all integers x.

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