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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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