“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.
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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).