2. The product of two three-digit numbers is a number written only with the digit 3.

What are those numbers?
Solution. The product of two three-digit numbers is greater than or equal to $100 \cdot 100=10000$, and less than $1000 \cdot 1000=1000000$. Therefore, the product of two three-digit numbers is a five-digit or six-digit number.

If the product is a five-digit number, then

$$
33333=3 \cdot 11111=3 \cdot 41 \cdot 271=123 \cdot 271
$$

which means the required numbers are 123 and 271.

If the product is a six-digit number, then

$$
333333=3 \cdot 111111=3 \cdot 111 \cdot 1001=3^{2} \cdot 7 \cdot 11 \cdot 13 \cdot 37
$$

Further, we have

$$
\begin{aligned}
& \left(3^{2} \cdot 7 \cdot 11\right)(13 \cdot 37)=693 \cdot 481 \\
& \left(3^{2} \cdot 7 \cdot 13\right)(11 \cdot 37)=819 \cdot 407 \text { and } \\
& (3 \cdot 7 \cdot 37)(3 \cdot 11 \cdot 13)=777 \cdot 429
\end{aligned}
$$

which means the required numbers are 481 and $693 ; 407$ and $819 ; 429$ and 777.