<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] Doing it by the book, though you were already told there are simpler/faster ways:
For $\;n=1:\;\;1^4-1^2=0\;\color{green}\checkmark\;$
Suppose it is true for $\;n\;$ and prove for $\;n+1\;$
$$(n+1)^4-(n+1)^2=(n+1)^2\left((n+1)^2-1\right)=(n+1)^2(n+1-1)(n+1+1)=$$
$$(n^2+2n+1)(n^2+2n)=n^4+4n^3+5n^2+2n=n^4-n^2+2n(2n^2+3n+1)=$$
$$=n^4-n^2+2n(n+1)(2n+1)$$
Now observe that the factor $\;2n(n+1)(2n+1)\;$ is always divisible by $\;4\;$ (as either $\;n\;$ or $\;n+1\;$ is even), and also by $\;3\;$ (since either $\;n\;$ is, or $\;n=1\pmod 3\;$ and then $\;2n+1=0\pmod 3\;$ , or $\;n=2\pmod3\;$ and then $\;n+1=0\pmod3\;$)
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