Eva thought of two natural numbers. She first correctly added the numbers, then subtracted them. In both cases, she got a two-digit result. The product of the two-digit numbers thus created was 645.

Which numbers did Eva think of?

(E. Novotná)

Hint. Every natural number has a finite number of divisors.
The number 645 can be written as the product of two natural numbers in the following ways:

$$
645=1 \cdot 645=5 \cdot 129=15 \cdot 43=3 \cdot 215
$$

The factors are two-digit numbers only in the last case, so the sum of Eva's numbers was 43 and the difference was 15.

The numbers 43 and 15 were obtained by adding and subtracting the smaller of Eva's numbers from the larger one. Thus, the larger of Eva's numbers is exactly between (i.e., the average of) 43 and 15, and the smaller of Eva's numbers is half the difference between 43 and 15. Eva was thinking of the numbers

$$
\frac{43+15}{2}=29, \quad \frac{43-15}{2}=14 .
$$

Note. If we denote the larger of Eva's numbers by $v$ and the smaller by $m$, then the previous relationships can be written as

$$
v+m=43, \quad v-m=15 .
$$

From this, we can formally derive that

$$
(v+m)+(v-m)=43+15, \quad(v+m)-(v-m)=43-15
$$

and thus

$$
v=\frac{43+15}{2}=29, \quad m=\frac{43-15}{2}=14
$$