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.

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Answered by Isaac H. Maths tutor

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