<s>[INST] Prove $30|(a^3b-ab^3) $ Prove that if three distinct integers are chosen at random then there will exists two among them, say $a$ and $b$ such that $30|(a^3 b-ab^3)$
 [/INST] By Fermat's little theorem (or simply by direct calculation), $x^2\equiv x$ mod $2$ and $x^3\equiv x$ mod $3$ for any integer $x$.  It follows that
$$a^3b\equiv ab\equiv ab^3\mod 6$$
for any pair of integers $a,b$, hence $6$ divides $a^3b-ab^3$ for any pair.  It remains to show that $5$ divides $a^3b-ab^3=ab(a^2-b^2)$ for at least one of the three pairs.  But there are only three squares mod $5$, namely $0$, $1$, and $-1$.  So given three integers, either two of their squares, say $a^2$ and $b^2$ belong to the same residue class, in which case $5$ divides $a^2-b^2$, or one of the three squares, say $a^2$ is $0$ mod $5$, in which case $5$ divides $a$ (and you can let $b$ be either of the other two numbers).  In either case $5$ divides $ab(a^2-b^2)=a^3b-ab^3$ for the chosen pair, and we are done.
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