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# HCF Expression

##### Age 11 to 14 ShortChallenge Level

Answer: $45$

$2p^2q=pq(2p)$ and $3pq^2=pq(3q)$ so $pq$ is a common factor

$p$ and $q$ are $2,3$ or $5$

$3$ and $5$ : $2\times3^2\times5$ and $3\times3\times5^2$ (HCF $3^2\times5=45$)
or $2\times5^2\times3$ and $3\times5\times3^2$ (HCF only $3\times5=15$)

$2$ and $5$ : won't be better because $2\lt3$ and effectively this swaps $2$ and $3$

$2$ and $3$ : $2\times3^2\times2$ and $3\times2^2\times3$ so both numbers are equal to $4\times9=36$

Best option was $3$ and $5$ to give HCF $45$

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.