In a meeting of mathematicians, Carlos says to Frederico: $O$ double the product of the two digits of the number of mathematicians in the meeting is exactly our quantity. What is the minimum number of mathematicians that must join us for our quantity to be a prime number? Help Frederico solve the problem.

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Solution

Let's represent the number of mathematicians by $\overline{a b}$. We have:

$$
\begin{aligned}
2 a b & =\overline{a b} \\
2 a b & =10 a+b \\
2 a b-b & =10 a \\
b(2 a-1) & =10 a \\
b & =\frac{10 a}{2 a-1}
\end{aligned}
$$

Analyzing the possible values of $a$, remembering that both are digits, with $a \neq 0$, we conclude that $a=3$ and $b=6$. Therefore, the number of mathematicians is 36, and with one more mathematician, the quantity will be a prime number.