4. The product of two three-digit numbers is a multi-digit number written using only the digit 3. What are those three-digit numbers?
Solution. The product of two three-digit numbers is greater than or equal to $100 \cdot 100=10000$, and less than $1000 \cdot 1000=1000000$, so it is clear that the product is either a five-digit or a six-digit number. If the product is a five-digit number, then

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

If the product is a six-digit number, then

$$
333333=3 \cdot 111111=3 \cdot 111 \cdot 1001=3 \cdot 3 \cdot 7 \cdot 11 \cdot 13 \cdot 37=777 \cdot 429
$$

Thus, the pairs of three-digit numbers are 123 and 271, 777 and 429.