Question

If $p$ and $q$ are any two positive integers, show that $|\underset{–}{pq}$ is divisible by $(|\underset{–}{p}{)}^q.|\underset{–}{q}$ and by $(|\underset{–}{q}{)}^p.|\underset{–}{p.}$

Answer 1

These results are easily established by Induction.

Suppose that, $|\underset{–}{p(q-1)}$ is divisible by ${\left(|\underset{–}{p}\right)}^{q-1}|\underset{–}{q-1}.$

Now $\frac{\left(pq\right)!}{{(p!)}^{q}q!}÷\frac{\left(p(q-1)\right)}{{(p!)}^{q-1}(q-1)!}=\frac{\left(pq\right)!}{(pq-p)!}\times \frac{(q-1)!}{p!q!}$

$=\frac{pq(pq-1)(pq-2)......pfactors}{pq(p-1)!}$

$=\frac{(pq-1)(pq-2).......(p-1)factors}{(p-1)!}$

But $\frac{p!}{p!1!}$ is an integer.

Hence, $\frac{\left(2p\right)!}{{(p!)}^{2}2!}$ is an integer; and so on.