Question

Let $N$ be the greatest integer multiple of $8,$ no two of whose digits are the same. What is the remainder when $N$ is divided by $1000?$

答案

Solution:

An integer is divisible by $8$ exactly when the number formed by its last three digits is. To make $N$ as large as possible, use all ten digits once each and put the largest digits first: the leading digits are $9876543,$ and the final three digits are some arrangement of $0,$ $1,$ $2$ — provided one of those arrangements is a multiple of $8.$

Checking $012,$ $021,$ $102,$ $120,$ $201,$ $210,$ the only multiple of $8$ is $120.$ So $N = 9{,}876{,}543{,}120,$ and the remainder upon division by $1000$ is $120.$

2003 AIME II Problem 2

Solution:

Problem 2 in Other Years