## Task 13/81

When asked about the ages of her three children, a mathematician replied: "The sum of their whole age numbers is our house number, the product of the same is 72, and to make the result unique, I also mention that my youngest child is a girl."

How old are the children? The questioner knew the house number.
Without regard for biological possibilities, the number 72 can be factored into three factors in the following ways:

$1 \cdot 1 \cdot 72, 1 \cdot 2 \cdot 36, 1 \cdot 3 \cdot 24, 1 \cdot 4 \cdot 18, 1 \cdot 6 \cdot 12, 1 \cdot 8 \cdot 9, 2 \cdot 2 \cdot 18, 2 \cdot 3 \cdot 12, 2 \cdot 4 \cdot 9, 2 \cdot 6 \cdot 6, 3 \cdot 3 \cdot 8, 3 \cdot 4 \cdot 6$. Since the questioner knew the house number, he could have determined the correct one from these 12 possibilities using the house number, if the house number did not appear as the sum of the factors more than once. In fact, the factorizations $2 \cdot 6 \cdot 6$ and $3 \cdot 3 \cdot 8$ both have the same factor sum of 14, so there is no uniqueness with this house number.

This ambiguity is resolved by the remark about the youngest child. The children are therefore 2 years old (a girl) and 6 years old (twins).