Will really appreciate some guidance.
Let $p$ be a prime and $n$ be a positive number.
Then $p^a$ exactly divides $n$ if $p^a|n$, but $p^{a+1} \not \! | \; n$. We then write $p^a\|n$ if $a$ is the largest component of $p$ such that $p^a|a$.
Prove that if $p^a\|n$ and $p^b\|m$ then $p^{a+b}\|mn$.