Use mathematical induction to prove that n3 n is divisible

Use mathematical induction to prove that n^3 - n is divisible by 3 for all positive integers n.

Solution

Let n = 1, n^3 - n = 0 ( which is divisble by 3 )
if n = k, then k^3 - k ( is divisible by 3 )
then n = ( k + 1 ); (k + 1 )^3 - ( k + 1 )
= k^3 + 3k^2 + 3k + 1 - ( k + 1 )
= (k^3 - k ) + 3k^2 + 3k
= ( k^3 - k ) + 3k(k + 1 )

here 3k(k + 1 ) is the multiple of 3. hence the whole thing is divisble by 3.