Question

Show that n4 − n3 + n2 − n is divisible by 2 for all positive integers n

Answer 1

The expression given to us is $n^4-n^3+n^2-n$

This can be simplified as:

$n(n^3-n^2+n-1)=n(n^2(n-1)+(n-1))=n(n^2-1)(n-1)$

Now, division by 2 will look like: $p=\frac{n(n^2-1)(n-1)}{2}$

It can be clearly seen that if n is even then the n at the front will easily be divisible by 2.

Likewise, when n is odd, the last expression n-1 will be even and thus that will be easily divisible by 2. Thus, in both the cases when n is even or odd, the given expression $n^4-n^3+n^2-n$ is seen to be divisible by 2 in both cases.