Problem:
Find the number of positive integers with three not necessarily distinct digits, , with such that both and are multiples of .
Solution:
Because any multiple of is even, it must be the case that both and are even. Furthermore, an integer is a multiple of if and only if the integer is a multiple of . Thus both and are multiples of . It follows that is a multiple of , so and leave the same remainder upon division by . Thus must be a subset of either or .
If or or or , then can be any of the digits , or , for a total of possibilities.
If or or or , then can be any of the digits , or , also for a total of possibilities.
Thus there are positive integers satisfying the conditions.
The problems on this page are the property of the MAA's American Mathematics Competitions