MathNet 2003 — Part 01 Question 47

We consider the natural numbers $n$ with three digits, all of them non-zero. We will say that a number $n$ is good if the number $n+1$ is a multiple of the two-digit number obtained by eliminating the first digit on the left from $n$ (that is, by eliminating the hundreds digit). For example, $123$ is NOT good, because $124$ is not a multiple of $23$.

Find all the good numbers.