Question

All of the digits of a three-digit integer are distinct and non-zero. Furthermore, the three-digit integer is divisible by 17. Find the largest three-digit integer that has these properties.

Answer 1

To make the division easy, we want to find a multiple of $17$ which is close to $900$. Trying some trials, we find that $17\cdot53=901$, so our number is $901$. However, we need our number to have distinct digits. To maintain divisibility by $17$, note that $34$ is a multiple of $17$, so the number $17\cdot((10\cdot3)+4)=906$ is a multiple of $17$ and uses distinct digits. So we must find multiples of $17$ which are less than $906$. Trying some more multiples of $17$, we find that $17\cdot(10\cdot5+1)=17\cdot51=867$, and so $\boxed{867}$ is the largest number which satisfies the condition.