Hcf expression
Find out which two distinct primes less than $7$ will give the largest highest common factor of these two expressions.
Age
11 to 14
Challenge level
Problem
If $p$ and $q$ are distinct primes less than $7$, what is the largest possible value of the highest common factor of $2p^2 q$ and $3pq^2$?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
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 $45$