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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.

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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?

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Factor Trio

Weekly Problem 45 - 2013
Which of the numbers shown is the product of exactly 3 distinct prime factors?

Three Primes

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

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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