Prove that n3 n is divisible by 3 for all integers nSolutio

Prove that n^3 - n is divisible by 3 for all integers n.

Solution

For n=1,
n^3n=11 which is divisible by 3

Assume the statement is true for some number n, that is, n^3n is divisible by 3.

Now,

(n+1)^3(n+1)=n^3+3n^3+3n+1n1=(n^3n)+3(n^2+n)

which is n^3n plus a multiple of 3.

Since we assumed that n^3n was a multiple of 3, it follows that (n+1)^3(n+1) is also a multiple of 3.

So, since the statement \"n^3n is divisible by 3\" is true for n=1, and its truth for n implies its truth for n+1, the statement is true for all whole number n.