The discussion revolves around proving that for any integer \( n \), the expression \( 4 | n (n^2 - 1) (n + 2) \) holds true. Participants explore the application of the quotient remainder theorem and the properties of consecutive integers in this context.
Participants generally agree on the approach of using the properties of consecutive integers to argue for divisibility by 4, but there is no consensus on how to formally prove the argument or on the next steps to take.
The discussion does not resolve the formal proof of the divisibility claim, and participants express varying levels of confidence in their reasoning.
Ackbach said:Couple of hints.
1. $n(n^2-1)(n+2)=(n-1)\, n \,(n+1)(n+2)$.
2. By the Quotient Remainder Theorem, there exist integers $q$ and $r$ such that $n=4q+r$, where $0\le r<4$.
Does this suggest anything to you?
tmt said:Right, so $(n-1)\, n \,(n+1)(n+2)$ is 4 consecutive integers. I get that if you take any arbitrary integer, if it is not divisible by 4, you can increment it by some int $c$ such that $0< c < 4$ and get an int that is divisible by 4.
Therefore, if you have 4 consecutive integers, one of those integers will be divisible by 4, and as a result the product of those 4 integers will be divisible by 4. I just don't know how to state or prove this formally.