The age of each of Paulo's three children is an integer. The sum of these three integers is 12 and their product is 30. What is the age of each of his three children?

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Solution

Let $a, b$ and $c$ be the ages of Paulo's three children. Then:

$$
\begin{aligned}
a+b+c & =12 \\
a b c & =30
\end{aligned}
$$

The set of divisors of 30 is $\{1,2,3,5,6,10,15,30\}$. Since they are positive integers and their sum is 12, it follows that $a, b, c \in\{1,2,3,5,6,10\}$. Since the product is 30, one of them must be a multiple of 5. If one of them is 10, for the product to be 30, the others can only be 1 and 3. This does not satisfy the condition of the sum of the ages. Therefore, one of the ages is 5. Thus, the product of the other two is $30 / 5=6$. The only possibilities are 2 and 3 or 1 and 6. The first one is not possible due to the condition of the sum of the ages. Therefore, the three ages are 1, 5, and 6.