Case-1 where n is odd-
now if n=2a+1(a is any natural no.),2 divides (n+1)(n-1) again (n-1),n and (n+1) are three consecutive numbers.Hence 3 divides at least one of the three numbers-n,(n+1) and (n-1). So,3*2 or 6 divides n^3-n
Case-2 where n is even-
if n=2a(a is any natural number),2 divides n.So,2 divides n^3-n.Again, by the same logic 3 divides n^3-n.So,3*2 or 6 divides n^3-n when n is even.
Thus proved that 6 divides n^3-n .