IMC 2009 Q18
18. If \(p\), \(q\) are distinct primes less than 7, what is the largest possible value of the highest common factor of \(2p^2q\) and \(3pq^2\)?
- A 60
- B 45
- C 36
- D 20
- E 15
Intermediate Mathematical Challenge 2009, Question 18. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 45
For all positive integer values of \(p\) and \(q\), \(2p^2q\) 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 \(q\), \(2p^2q\) and \(3pq^2\) have values 90 and 225 respectively, giving a highest common factor of 45.
Solution © UK Mathematics Trust, from the IMC 2009 solutions.