SOLUTION: Prove that for any integer n, {{{ n(n^2-1)(n+2) }}} is divisible by 12. Below is my attempt at trying to solve it, however I don't think example is enough to show as proof. w

Algebra ->  Proofs -> SOLUTION: Prove that for any integer n, {{{ n(n^2-1)(n+2) }}} is divisible by 12. Below is my attempt at trying to solve it, however I don't think example is enough to show as proof. w      Log On


   

Question 1070285: Prove that for any integer n, +n%28n%5E2-1%29%28n%2B2%29+ is divisible by 12.
Below is my attempt at trying to solve it, however I don't think example is enough to show as proof.
we want to prove +%28+n%28n%5E2-1%29%28n%2B2%29+%29+%2F+12+. I written it also as
+%28%28+%28n-1%29n%28n%2B1%29%28n%2B2%29%29+%2F12%29+ which if you plug in 2 or 3 it will give an even number in the numerator which means it can be divisible by 12.
Answer by ikleyn(52779) About Me  (Show Source):
You can put this solution on YOUR website!
.
You got this product   +%28%28+%28n-1%29n%28n%2B1%29%28n%2B2%29%29+%2F12%29+.

The numerator is the product of 4 consecutive integers,  isn't it ?

One of them is a multiple of 4.  One of them is a multiple of 3.

Therefore,  the numerator is a multiple of  12,

so the fraction is an integer number.

Proved.


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MOREOVER, it is a multiple of 24.