SOLUTION: Prove that for all integers n, that the last digit of n^5 is the same as the last digit of n.

Algebra ->  Exponents-negative-and-fractional -> SOLUTION: Prove that for all integers n, that the last digit of n^5 is the same as the last digit of n.      Log On


   

Question 594344: Prove that for all integers n, that the last digit of n^5 is the same as the last digit of n.
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
Just check ten cases, where each case covers a residue class (mod 10). In other words, we can simply check n = 0, 1, ..., 9 and we are done.

0^5 = 0
1^5 = 1
2^5 = 32 ≡ 2
3^5 = 243 ≡ 3
.
.
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9^5 = 59049 ≡ 9

You can fill in the remaining exponents if you wish. Point is, n^5 is congruent to n (mod 10), so they must have the same last digit.