Example 6 Find three prime numbers such that their product is 5 times their sum.

 

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Analysis Let the three prime numbers be represented by $p, q, r$, and using the properties of prime numbers, it is easy to solve for $p, q, r$.
Solution Let the three prime numbers be $p, q, r$,
then $p q r=5(p+q+r)$,
thus one of $p, q, r$ must be equal to 5. Without loss of generality, let $r=5$,
then $5 p q=5(p+q+5)$,
which simplifies to $p q=p+q+5,(p-1)(q-1)=6$.
Given $p>1, q>1$,
we have
$\left\{\begin{array}{l}p-1=2 \\ q-1=3\end{array}\right.$ or $\left\{\begin{array}{l}p-1=3 \\ q-1=2\end{array}\right.$ or $\left\{\begin{array}{l}p-1=1 \\ q-1=6\end{array}\right.$ or $\left\{\begin{array}{l}p-1=6 \\ q-1=1\end{array}\right.$
Since $p=4$ or $q=4$ is not a prime number,
the only solutions are $p=2, q=7$ or $p=7, q=2$.
Therefore, the prime numbers are $2,5,7$.