### 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.

### Factor Lines

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

### Stars

Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

# Three Primes

##### Stage: 3 Short Challenge Level:

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.
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