Problem:
Let be the greatest integer multiple of , no two of whose digits are the same. What is the remainder when is divided by ?
Solution:
An integer is divisible by if and only if the number formed by the rightmost three digits is divisible by . The greatest integer with the desired property is formed by choosing as the seven leftmost digits and finding the arrangement of that yields the greatest multiple of , assuming that such an arrangement exists. Checking the permutations of yields as the sole multiple of , so , and its remainder when divided by is .
The problems on this page are the property of the MAA's American Mathematics Competitions