You may also like

problem icon

Prompt Cards

These two group activities use mathematical reasoning - one is numerical, one geometric.

problem icon

Round and Round the Circle

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

problem icon

Strange Numbers

All strange numbers are prime. Every one digit prime number is strange and a number of two or more digits is strange if and only if so are the two numbers obtained from it by omitting either its first or its last digit. Find all strange numbers.

Weekly Problem 6 - 2010

Stage: 2 and 3 Short Challenge Level: Challenge Level:1

Let $p$, $q$ and $r$ be three prime numbers such that $pqr=5(p+q+r)$. Then one of the prime numbers must be $5$, say $r$.

This implies that $5pq=5(p+q+5)\Rightarrow pq=p+q+5\Rightarrow pq-p-q+1=6\Rightarrow (p-1)(q-1)=6$.

Therefore either $p-1=1$ and so $q-1=6$ i.e. $(p,q)=(2,7)$ (or vice versa) or $p-1=2$ and so $q-1=3$ i.e. $(p,q)=(3,4)$ (or vice versa). But $4$ is not prime, so the only triple of primes which satisfies the condition is $(2,5,7)$.

This problem is taken from the UKMT Mathematical Challenges.

View the previous week's solution
View the current weekly problem