Proof using quotient remainder theorem: 4 | n(n^2-1)(n+2)

[maeveoneill]

How would I undergo writing a proof of the statement "For any integer n, n(n^2 -1)(n+2) is divisible by 4.

The quotient raminder theorem states that given any integer n, and positive integer d, there exist unique and integers q and r such that n=dq +r and o</=r<d. In this case if we let entire function become n, then n=4q.

What do I have to do to proove that it is true??

thANK YOU

 

[royhaas]

Re: Proof using quotient remainder theorem

Note that you can write this as \(\displaystyle (n-1)n(n+1)(n+2)\).

 

[chivox]

Re: Proof using quotient remainder theorem

It can also be written as

(n^3 - n) (n + 2)
n^4 - n^2 + 2 n^3 - 2n

By induction, (n^4 - n^2) is divisible by 4 for all integer n.

Therefore, you could reduce the problem to proving that the quantity (2 n^3 - 2n) is divisible by 4 for all integer n.

Either one of these approaches will get you to the proof, using the theorem you specified.