Skip over navigation
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 $45$
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.
You may also like
Consecutive Numbers
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Days and Dates
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...