Problem 14 – March 1 – Circular Numbers

“Circular” Numbers?

This problem is from Clayton Dodge.

Prove that for any positive integer n, the number

n²(n²-1)(n²-4) is divisible by 360.

Submit your answers to mathpotw@acu.edu.  Details for submissions can be found here.

Solution

Correct solutions were submitted by:   Wyatt Witemeyer

The expression n²(n²-1)(n²-4) can be factored into (n-2)(n-1)n²(n+1)(n+2). Observe that in any set of five consecutive integers, exactly one of them is divisible by 5. Similarly, in a set of four consecutive integers, exactly one is divisible by 4, and another one (two units away) is divisible by 2. So far we have seen that the product in question is divisible by 5*8 = 40.

Now look at the product as ((n-2)(n-1)n)*(n(n+1)(n+2)). If 3 divides n-2, then 3 also divides n+1. If 3 divides n-1, then 3 also divides n+2. Finally, if 3 divides n, then 9 divides n². Exactly one of these three cases holds, so we see that 9*40 = 360 divides n²(n²-1)(n²-4).