<s>[INST] Proving that $ 30 \mid ab(a^2+b^2)(a^2-b^2)$ How can I prove that $30 \mid ab(a^2+b^2)(a^2-b^2)$ without using $a,b$ congruent modulo $5$ and then 
$a,b$ congruent modulo $6$ (for example) to show respectively that $5 \mid ab(a^2+b^2)(a^2-b^2)$ and 
$6 \mid ab(a^2+b^2)(a^2-b^2)$? 
Indeed this method implies studying numerous congruences and is quite long.
 [/INST] Hint for the divisors 2 and 3.
Note the factorisation $ab(a+b)(a-b)(a^2+b^2)$ which is divisible by $ab(a-b)(a+b)$
Either one of $a,b$ is divisible by 2 (or 3) or not. If not, look at the other factors to complete the proof.
Studying cases does not look too onerous to me.
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