Prove that n^5 - n is divisible by 30

proof:
we have n^5 - n = n(n^4-1) = n(n^2-1)(n^2+1) = n(n+1)(n-1) (n^2+1)

as n(n+1)(n-1) is product of 3 consecutive numbers it is divisible by 6

now under mod 5
n^2 + 1 = n^2- 4 = (n+2)(n-2)
so n^5 - n = n(n+1)(n-1)(n+2)(n-2)

as it is product of 5 consecutive numbers it is divisible by 5

as it is divisible by 5 and 6 hence 30

proved