## Task A-1.3. (4 points)

Marko is celebrating his birthday today. His father Joško and grandfather Luka are talking:

- Now, both Marko's and your and my age are prime numbers!
- Yes, and in five years, all three will be squares of natural numbers.

How old was Luka on the day of Marko's birth?
## Solution.

In five years, the ages of Markov, Joskov, and Lukin will be among the numbers

$$
1,4,9,16,25,36,49,64,81,100,121, \ldots
$$

which means that their current ages are among the numbers

$$
>4, \neg 1,4,11,20,31,44,59,76,95,116, \ldots
$$

Among these, the only prime numbers are 11, 31, and 59, so it is clear that

Marko is 11, Josko is 31, and Luka is 59 years old.

Therefore, on the day of Marko's birth, Luka was 48 years old.