 # Practice Paper 3 Question 2

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

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## Hints

• Hint 1
How about factorizing the expression.
• Hint 2
How can one show divisibility by $$12$$ by showing divisibility by smaller numbers?
• Hint 3
Would splitting the problem into cases help prove divisibility by $$4?$$
• Hint 4
... such as odd and even?

## Solution

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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