<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] 1) Divisibility by $5$.
Let $\{a,b,c\}$ the three numbers reduced modulo $5$. If one of them is $0$ or two of them are equal we clearly have
$$ab(a+b)(a-b)\equiv 0\pmod 5$$
So be all the three, $a,b,c$ distinct and non-zero modulo $5$ so we have
$$\{a,b,c\}=\{1,2,3\},\{1,2,4\},\{1,3,4\},\{2,3,4\}$$
In the four cases we clearly have two of the (class) numbers have sum $0$ modulo $5$ (because of $2+3=1+4$) so $ab(a+b)(a-b)\equiv 0\pmod 5$.
Thus the expression is for two of the three integers divisible by $5$.
2) Divisibility by $3$ and by $2$.
The same method but easier than for divisibility by $5$ can be used for divisibility by $3$ and by $2$.
Thus the divisibility by $30=2\cdot3\cdot5$ of the expression for two of the three numbers..
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