The Computer Laboratory

Practice Paper 3 Question 2

Show that \(12\) divides \(n^4 - n^2\) for all positive integers \(n.\)

The above links are provided as is. They are not affiliated with the Climb Foundation unless otherwise specified.


  • 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?


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.

If you have queries or suggestions about the content on this page or the CSAT Practice Platform then you can write to us at oi.footasc@sulp.ecitcarp. Please do not write to this address regarding general admissions or course queries.