## Task 2 - 310732

A person answers the question about their birthday:

"In the year 1989, I was $a$ years old. I was born on the $t$-th day of the $m$-th month of the year $(1900+j)$. The numbers $a, j, m, t$ are natural numbers; for them, $a \cdot j \cdot m \cdot t=105792.$"

Determine whether the numbers $a, j, m, t$ are uniquely determined by this information! If this is the case, provide these numbers!
From the meaning of the numbers, it follows that $0 < m \leq 12, 0 < t \leq 31$; (1) also, the numbers $a$ and $j$, for which $a \cdot j \cdot m \cdot t = 105792$ (2) holds, are consequently both greater than zero.

They further satisfy $a + j = 89$ (3) and are therefore both less than 89.

Due to the prime factorization $105792 = 26 \cdot 3 \cdot 19 \cdot 29$, 105792 has exactly the following divisors among the natural numbers less than 89: $1, 2, 3, 4, 6, 8, 12, 16, 19, 24, 29, 32, 38, 48, 57, 58, 64, 76, 87$.

The only possibilities to select two numbers $a, j$ from these that satisfy (3) are

$$
(a; j) = (2; 87), (87; 2), (32; 57), (57, 32). (4)
$$

From them, $(2; 87)$ and $(87, 2)$ are excluded; because due to (1), for them $a \cdot j \cdot m \cdot t \leq 2 \cdot 87 \cdot 12 \cdot 31 < 105792$, which contradicts (2).

Therefore, it follows from (2) that $m$ and $t$ must satisfy the condition

$$
m \cdot t = \frac{105792}{32 \cdot 57} = 2 \cdot 29
$$

Since 2 and 29 are prime numbers, this is only possible with $m = 2, t = 29$ (5) due to (1); i.e., the birthday can only have been February 29. This date is not possible with $j = 57$, i.e., for the year 1957, as it was not a leap year.

Thus, the only remaining possibility from (4) is $a = 57, j = 32$. (6)

This shows: The numbers $a, j, m, t$ are uniquely determined by the given information; they are as stated in (5) and (6).