Prove that for any two integers $a$ and $b$, $ab(a^{60}-b^{60}) $ is divisible by $56786730$
$56786730=2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot31\cdot61$
It is easy to prove that $2,31,61|\ ab(a^{60}-b^{60}) $ using Fermat's little theorem for 61 and 31 and taking cases for 2. But I can't figure out a way to solve this for all the other factors. Can anybody help me solve this.
Except this, I would also appreciate if someone could share another method to solve this question which doesn't involve modular congruence as I haven't really studied that yet.