Problem:
A number is called flippy if its digits alternate between two distinct digits. For example, and are flippy, but and are not. How many five-digit flippy numbers are divisible by
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Solution:
A five-digit flippy number looks like for some distinct digits and with (otherwise it would not be a five-digit number). In order for this number to be divisible by , it must be divisible by and .
Any number divisible by will have a units digit of or . Because , must equal . This means the flippy number must have the form .
For a number to be divisible by , the sum of its digits must be divisible by . The sum of the digits of is . This means must be divisible by , so is divisible by . There are possible digits for , or . Therefore there are five-digit flippy numbers divisible by , and .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions