## Task 13/67

Two brothers have their birthdays on the same day. The last time they celebrated, their combined age was 48 years. The age of one of them was indicated by a cubic number. If you divided his age and the age of the other by the base of this cubic number, the results in this order gave the date of the birthday.

How old were the two brothers and on which day did they have their birthday?
The only cube numbers that come into question are 1, 8, and 27, as the fourth cube number 64 is already greater than 48. If we try them in sequence, the following calculations result:

1. $48-1=47 ; 1: 1=1 ; 47: 1=47$

The date 1.47. is impossible (and it is also unlikely that one of the two brothers is 47 years old and the other is 1 year old).
2. $48-8=40 ; 8: 2=4 ; 40: 2=20$

The same applies to the result as under 1.
3. $48-27=21 ; 27: 3=9 ; 21: 3=7$

This results in the birth date of 9.7., so the two brothers were 27 and 21 years old, respectively.

Since the case distinction is complete, this result is unique.