## Task 1 - 090831

The age specifications (expressed in full years of life) of a family - father, mother, and their two children - have the following properties:

The product of all four ages is 44950; the father is 2 years older than the mother.

How old are the four family members?
The prime factorization of 44950 is $44950=2 \cdot 5 \cdot 5 \cdot 29 \cdot 31$. Therefore, there are exactly the following possibilities to decompose 44950 into a product of exactly 4 natural numbers:

(1) $(2 \cdot 5) \cdot 5 \cdot 29 \cdot 31=10 \cdot 5 \cdot 29 \cdot 31$,

(2) $(2 \cdot 29) \cdot 5 \cdot 5 \cdot 31=58 \cdot 5 \cdot 5 \cdot 31$,

(3) $(2 \cdot 31) \cdot 5 \cdot 5 \cdot 29=62 \cdot 5 \cdot 5 \cdot 29$,

(4) $(5 \cdot 5) \cdot 2 \cdot 29 \cdot 31=25 \cdot 2 \cdot 29 \cdot 31$,

(5) $(5 \cdot 29) \cdot 2 \cdot 5 \cdot 31=145 \cdot 2 \cdot 5 \cdot 31$,

(6) $(5 \cdot 31) \cdot 2 \cdot 5 \cdot 29=155 \cdot 2 \cdot 5 \cdot 29$,

(7) $(29 \cdot 31) \cdot 2 \cdot 5 \cdot 5=899 \cdot 2 \cdot 5 \cdot 5$

Since the age difference between the two parents is 2 years, only cases (1) and (4) can be solutions. Of these, case (4) is not realistic; according to it, the 29-year-old mother would have a 25-year-old child.

Thus, only possibility (1) remains; i.e., the sought age specifications can only be 31, 29, 10, 5. Conversely, these specifications indeed meet all the conditions of the problem.