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
.