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x^2is a answer of your question
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You were on the right track initially.
n3−n=n(n2−1)=n(n+1)(n−1).
These are three consecutive integers, so at least one of them is even. Additionally, 3 must divide one of the numbers by the same logic. Thus n3−n is divisible by 6 since it is divisible by 2 and 3.
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