Show that $12$ divides $n^4 - n^2$ for all positive integers $n.$

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How about factorizing the expression.

How can one show divisibility by $12$ by showing divisibility by smaller numbers?

Would splitting the problem into cases help prove divisibility by $4?$

… such as odd and even?

Factorise to get $n^4-n^2$ $=n^2(n^2-1)$ $=n^2(n+1)(n-1)$ and we’ll show that it’s divisible by both 3 and 4. As $n-1, n,n+1$ are three consecutive integers, 3 must divide the expression. To prove that 4 divides the expression, we will consider two cases. If $n$ is even then $n^2$ is divisible by 4. If $n$ is odd then both $n+1$ and $n-1$ are even, hence their product is divisible by 4.

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