2017 AIME I Problems/Problem 9
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Problem 9
Let , and for each integer let . Find the least such that is a multiple of .
Solution
Writing out the recursive statement for and summing them gives Which simplifies to Therefore, is divisible by if and only if is divisible by 99, so needs to be divisible by 9 and 11. Assume that is a multiple of 11. Writing out a few terms, , we see that is the smallest that works in this case. Next, assume that is a multiple of 11. Writing out a few terms, , we see that is the smallest that works in this case. The smallest is .