2024 AMC 10A Problem 10

Doing $100$ steps by hand is too many, but the numbers might start repeating. So do the first few steps and watch.

1. Start at $100$. It is not a multiple of $3$, so add $10$: $110$. Keep going:

(For example, $120$ is a multiple of $3$, so it becomes $40$; $60$ becomes $20$; $30$ becomes $10$; $10$ is not a multiple of $3$, so it becomes $20$.)
2. At step $9$ we are back at $20$, the value from step $6$. From then on the same three numbers repeat forever: $20, 30, 10, 20, 30, 10, \ldots$
3. Match each step with its value:
- steps $6, 9, 12, \ldots$ (multiples of $3$) give $20$,
- steps $7, 10, 13, \ldots$ (one more than a multiple of $3$) give $30$,
- steps $8, 11, 14, \ldots$ (two more than a multiple of $3$) give $10$.
4. $100 = 99 + 1$, and $99 = 3 \cdot 33$. So $100$ is one more than a multiple of $3$, just like step $7$. The value is $30$.

The answer is $\boxed{\textbf{(C)}\ 30}$.

A rule that always does the same thing to the same number must start repeating as soon as a number comes back. Work out steps until you see a repeat, note how long the repeating block is, and then use division to find where the target step lands in the block. Line it up using a step you have actually computed, so you do not end up one step off.