3. Polycarp instead of the usual multiplication of three-digit numbers decided to simply "glue the numbers", appending one number to the other. The result turned out to be 7 times larger than the usual. What numbers was Polycarp multiplying?
Answer: 143 and 143.

Solution. Let the original numbers be denoted by $a$ and $b$. The result of the usual multiplication is $a b$, and the result of the "concatenation" is $-1000 a + b$. According to the condition, they are related by the equation $1000 a + b = 7 a b$, from which we get $b = \frac{1000 a}{7 a - 1}$. Note that the numbers $a$ and $7 a - 1$ have no common divisors, so $p = 7 a - 1$ is a divisor of 1000. Given that $a \geqslant 100$, we have $p \geqslant 699$, but then $p = 1000, a = \frac{1000}{7} = 143$ and $b = a$.

Criteria. Only the answer - 0 points, the equation is correctly set up - 3 points. The correct answer is derived, but it is not proven that there are no others - 5 points.