2024 AMC 10A Problem 10

Consider the following operation. Given a positive integer

n

, if

n

is a multiple of

3

, then you replace

n

\frac{n}{3}

. If

n

is not a multiple of

3

, then you replace

n

n+10

. For example, beginning with

n=4

, this procedure gives

4\to14\to24\to8\to18\to6\to2\to12\to\cdots

. Suppose you start with

n=100

. What value results if you perform this operation exactly

100

times?

\text{(A)}\;10

\text{(B)}\;20

\text{(C)}\;30

\text{(D)}\;40

\text{(E)}\;50