1003. A mathematician was asked: "How old are you?" He answered in a complicated way: "The number of my years has an odd number of distinct divisors, and the number of these divisors also has an odd number of distinct divisors." So how old is he?
$\triangle$ Based on the statement of problem 1001, the number of years of the mathematician must be a perfect square. Let's list all such numbers:

$$
4,9,16,25,36,49,64,81,100,121,144,169
$$

(it is probably unrealistic to go further). Let's count the number of divisors of each of these numbers, using the statement of problem 995. We will get the following respectively:

$$
3,3,5,3,9,3,7,5,9,3,15,3
$$

Among them, in turn, we will find the numbers that are perfect squares. There is only one such number - 9. It appears in two places, so the number of years of the mathematician takes two values, which can be found in the corresponding places in the first row.

Answer: 36 or 100.