Three primes

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 $(2,5,7)$.