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
Answer Choices:
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Solution:
For a number to be divisible by , the number must be divisible by both and . First, to ensure the number is divisible by , it must end in or in . Since the digits alternate between two distinct digits, we call the other digit . This would make our number either or . We can ignore the numbers in the form as they would reduce to a -digit number. As such, we know our number is for some digit . To ensure our number is also a multiple of , the sum of the digits must be a multiple of . The sum of our digits is:
Since is a multiple of , all that is required is that is a multiple of . This means can be , or . Therefore, we have 4 solutions. Thus, the correct answer is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions