## 40. Question about age

One fine spring Sunday morning, the head of the family went for a walk with his sons.

- Have you noticed,- he said to them,- that the age of the oldest of you is equal to the sum of the ages of the other two brothers?
- Yes. And we also noticed,一 they replied in unison,一 that the product of our ages and your age is equal to the sum of the cube of the number of your sons, multiplied by a thousand, and ten times the square of this number.

Could you determine from this conversation the age of the head of the family at the time of the birth of the second son?
40. The head of the family has three sons. The product of the ages of the father and all his sons is

$$
3^{3} \cdot 1000 + 3^{2} \cdot 10 = 27090 = 43 \cdot 7 \cdot 5 \cdot 3^{2} \cdot 2
$$

Since the age of the oldest son is equal to the sum of the ages of his brothers, the only possible factorization of the obtained number is

$$
43 \cdot 14 \cdot 9 \cdot 5
$$

Thus, the father is 43 years old, the second son is 9 years old. Therefore, when the middle son was born, the father was 34 years old.