8-5. Diligent Asey multiplied two three-digit numbers, while lazy Petya simply wrote them down one after the other. Petya's result turned out to be 7 times larger than Asey's. What numbers did she multiply?
Answer: 143 and 143

Solution. Let the original numbers be denoted by $a$ and $b$. Asey's result is $a b$, and Petya's result is $-1000 a + b$. 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 $7 a - 1 = p$ is a divisor of 1000. Given that $a \geq 100$, we have $p \geq 699$, but then $p = 1000, a = 1001 / 7 = 143, b = a$.

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.