Problem 5 · Three primes, product and sum
One of the primes must be 5, and $(q-1)(r-1) = 6$ does the rest.
Integer answer, at most 4 digitsHow many triples of prime numbers satisfy the condition that their product is 5 times their sum?
Copa Cangur · SCM
Medium
Closed answer
Reasoned solution
Key idea: if $pqr = 5(p+q+r)$, the product is a multiple of $5$; since $p$, $q$, $r$ are primes, one of them is exactly $5$.
Set $p = 5$: the equation becomes $5qr = 5(5 + q + r)$, i.e. $qr = 5 + q + r$. Rearrange to factor:
$$qr - q - r + 1 = 6 \;\Longleftrightarrow\; (q-1)(r-1) = 6.$$
The factorisations of $6$: $1 \cdot 6$ gives $(q, r) = (2, 7)$ — both prime ✓; $2 \cdot 3$ gives $(q, r) = (3, 4)$ — and $4$ is not prime ✗.
The only triple is $\{2, 5, 7\}$: product $70 = 5 \cdot 14 = 5(2+5+7)$ ✓.
Answer: 1