Use mathematical induction to show that (n to the power 5) - n is divisible by 5 for every positive integer n.
Sorry again for the expression. Thanks guys
n = 1
1^5 - 1 = 1 - 1 = 0
Assume true for n = k
k^5 - k = 5m for some integer m
Suppose n = k + 1
(k + 1)^5 - (k + 1) = k^5 + 5k^4 + 10k^3 + 10k^2 + 5k + 1 - k - 1
= k^5 - k + 5(k^4 + 2k^3 + 2k^2 + k)
= 5m + 5(k^4 + 2k^3 + 2k^2 + k)
= 5(m + k^4 + 2k^3 + 2k^2 + k)
Hence true for n = k + 1 and by the principle of mathematical induction true for all natural numbers n