For all positive integer values of $p$ and $q$, $2p^2 q$ and
$3pq^2$ have a common factor of $pq$.
They will also have an additional common factor of $2$ if $q =2$
and an additional common factor of $3$ if $p=3$.
As the values of $p$ and $q$ are to be chosen from $2, 3$ and $5$,
the largest possible value of the highest common factor
will occur when $p=3$ and $q=5$.
For these values of $p$ and $q$, $2p^2 q$ and $3pq^2$ have values
$90$ and $225$ respectively, giving a highest common factor of
This problem is taken from the UKMT Mathematical Challenges.