## Task 16/69

Two friends A and B meet. When asked about the age of his children, A says: "You can figure it out yourself. My four children are a total of 15 years old. The product of their integer age values is equal to your postal code."

B calculates. After a while, he asks: "Are there twins among your children?" A answers, and immediately B states the ages of the children.

What was A's answer? How old are the children?
B formed all quadruples of natural numbers whose sum is 13, where the order does not matter and repetition is allowed. For each quadruple, he calculated the product of the four numbers. If his postal code had appeared as a product only once, he could have immediately given the answer.

In reconstructing B's thought process, we can thus exclude all quadruples whose product appears only once. The following quadruples then remain:

Product $40: \quad(1 ; 1 ; 5 ; 8),(1 ; 2 ; 2 ; 10)$

Product $72: \quad(1 ; 2 ; 6 ; 6),(1 ; 3 ; 3 ; 8),(2 ; 2 ; 2 ; 9)$

Product $96: \quad(1 ; 4 ; 4 ; 6),(2 ; 2 ; 3 ; 8)$

All quadruples except $(2 ; 2 ; 2 ; 9)$ contain exactly 2 identical numbers. If A had answered "Yes" to B's question, B would not have been able to name the ages of the children. The answer was therefore "No," so B gave the ages of the children as 2, 2, 2 (triplets) and 9.