## Task 12/76

Mrs. Quidam tells: "My husband, I, and our four children all have the same birthday. On our last birthday, we added up our ages and got our house number. When we multiplied them, we got the mileage on our Trabant before the last major overhaul: $180523.$"

How old are the six Quidams? What is their house number?
To find the ages, the product 180523 is first factored into its prime factors: $180523=$ $7 \cdot 17 \cdot 37 \cdot 41$.

It is noted that there are only four factors. Therefore, two more factors need to be added; due to the requirement of integer values, these two factors can only be 1: $180523=1 \cdot 1 \cdot 7 \cdot 17 \cdot 37 \cdot 41$. Thus, Quidam has twins aged 1 year, as well as children aged 7 and 17 years, and the spouses are 37 and 41 years old.

Other mathematically possible solutions, such as $180523=1 \cdot 1 \cdot 1 \cdot 119 \cdot 37 \cdot 41$, are biologically implausible. The house number is thus $1+1+7+17+37+41=104$.