All of the digits of a three-digit integer are distinct and non-zero. Furthermore, the three-digit integer is divisible by 19. Find the largest three-digit integer that has these properties.

Answer 1

We first list out possible three-digit multiples of $19$. Because the remainder of $100\div19$ is $5$, the digits of multiples of $19$ must be between $055$ and $094$ inclusive. After manually checking divisibility, we find that the only three-digit numbers that fit this condition are $095, 114, 133, 152, 171, 190, 209, 228, 247, 266, 285, 304, 323, 342, 361, 380$. Notice that if we reverse the order of each of these numbers, we go from a relatively small number to relatively an even greater number, so we try reversing $\boxed{380}$, which satisfies all the given conditions.