Problem:
Consider all positive four-digit integers consisting of only even digits. What fraction of these integers are divisible by
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Note that a four-digit integer is divisible by if and only if the integer formed by cutting off the last two digits is divisible by . Since all the digits must be even, the last two digits must be one of , or , which totals to different possibilities. Since the leading digit cannot equal zero, there are a total of different four-digit integers that are divisible by and satisfying this condition. Noting that there are different possible four-digit integers that have all even digits, our final answer comes to be .
The problems on this page are the property of the MAA's American Mathematics Competitions