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'Three Primes' printed from https://nrich.maths.org/
Answer: 2, 5 and 7
Searching for possibilities
Product is $5\times$sum, so product is a multiple of $5$
$\therefore$ one of the numbers is $5$
$5\times\underline{ }\times\underline{ }=5\times(5+\underline{ }+\underline{ })$
$\Rightarrow \underline{ }\times\underline{ }=5+\underline{ }+\underline{ }$
$3\times7=21$ $5+3+7=15$ numbers too big
$3\times5=15$ $5+3+5=13$ numbers too small
$5\times5=25$ $5+5+5=15$ numbers too big
$2\times7=14$ $5+2+7=14$ yes
All reasonable possibilities tested
Using algebra
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 $(2,5,7)$.