The discussion centers on proving that for any integer \( n \), the expression \( 4 | n(n^2 - 1)(n + 2) \) holds true. The participants utilize the Quotient Remainder Theorem, which states that any integer \( n \) can be expressed as \( n = 4q + r \) where \( 0 \leq r < 4 \). They conclude that the product \( (n-1)n(n+1)(n+2) \) consists of four consecutive integers, ensuring that at least one of these integers is divisible by 4, thus confirming the original statement.
PREREQUISITESMathematics students, educators, and anyone interested in number theory, particularly those looking to understand proofs involving divisibility and integer properties.
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.