Prove n^3 - n is a multiple of 3

To prove n³-n is a multiple of 3 we rely on a few simple tricks. The first is to factorise the expression.n³-n = n(n²-1)n(n²-1) = (n-1)(n)(n+1)The next trick is to realise that the series of numbers n-1, n, n+1 are consecutive. For example if n = 2:n-1 = 1n = 2n+1 = 3If you have a series of 3 consecutive numbers, clearly one of them will be a multiple of 3. Hence if; n³-n = (n-1)(n)(n+1), for all n and one of the numbers n-1, n, n+1 is a multiple of 3, then n³-n is also a multiple of 3.