<s>[INST] Divisibility of $n^4 -n^2$ by 4 (induction proof) We have to show that $$ n^4 -n^2 $$  is divisible by 3 and 4 by mathematical induction  
Proving the first case is easy however I do not  know how what to do in the inductive step.  Thank you.
 [/INST] Let's prove that
if $n^4-n^2$ is a multiple of both $3$ and $4$,
then so is $(n+1)^4-(n+1)^2$.
Consider 
$
((n+1)^4-(n+1)^2)-(n^4-n^2)=4 n^3+6 n^2+2 n
$.
Ignoring $6n^2$, we have $4 n^3+2 n=3n^3+3n+n^3-n=3n^3+3n+6\binom{n+1}{3}$, which is always a multiple of $3$.
Ignoring $4n^3$, we have
$6 n^2+2 n=4n^2+2n(n+1)=4n^2+4\binom{n+1}{2}$, which is always a multiple of $4$.
We're done.
This problem is definitely one which is much easier not by induction but instead simply by arguing that $n^{4} - n^{2} = n^{2} (n + 1) (n - 1)$, as @eltonjohn suggested.
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