Challenge Level

As the product of each pair has the same value, this value must be product of the smallest and largest numbers, that is $5\times72$.

So the number which is paired with $10$ is $5\times72\div10$, that is $36$.

Only four have a factor of $5$ so each pair must contain one of these

$5$ has no factors of $2$ or $3$, so pairs with $72$, which has the highest powers of $2$ and $3$

To get $2^3$ and $3^2$, $10$ pairs with $36$

The others can also be paired up:

This problem is taken from the UKMT Mathematical Challenges.

You can find more short problems, arranged by curriculum topic, in our short problems collection.