AMC10 2024 A

AMC10 2024 A · Q10

AMC10 2024 A · Q10. It mainly tests Remainders & modular arithmetic, Sequences in number theory (remainders patterns).

Consider the following operation. Given a positive integer $n$, if $n$ is a multiple of $3$, then you replace $n$ by $\frac{n}{3}$. If $n$ is not a multiple of $3$, then you replace $n$ by $n+10$. Then continue this process. For example, beginning with $n=4$, this procedure gives $4 \to 14 \to 24 \to 8 \to 18 \to 6 \to 2 \to 12 \to \cdots$. Suppose you start with $n=100$. What value results if you perform this operation exactly $100$ times?

考虑以下操作。给定一个正整数 $n$，如果 $n$ 是 $3$ 的倍数，则用 $\frac{n}{3}$ 替换 $n$；如果 $n$ 不是 $3$ 的倍数，则用 $n+10$ 替换 $n$。然后继续此过程。例如，从 $n=4$ 开始，此过程得到 $4 \to 14 \to 24 \to 8 \to 18 \to 6 \to 2 \to 12 \to \cdots$。假如从 $n=100$ 开始，进行恰好 $100$ 次此操作，结果是多少？

(A) 10 10

(B) 20 20

(D) 40 40

(E) 50 50

Answer

Correct choice: (C)

正确答案：（C）

Solution

Let $s$ be the number of times the operation is performed. Notice the sequence goes $100 \to 110 \to 120 \to 40 \to 50 \to 60 \to 20 \to 30 \to 10 \to 20 \to \cdots$. Thus, for $s \equiv 1 \pmod{3}$, the value is $30$. Since $100 \equiv 1 \pmod{3}$, the answer is $\boxed{\textbf{(C) }30}$.

设 $s$ 为操作次数。注意序列为 $100 \to 110 \to 120 \to 40 \to 50 \to 60 \to 20 \to 30 \to 10 \to 20 \to \cdots$。因此，当 $s \equiv 1 \pmod{3}$ 时，值为 $30$。由于 $100 \equiv 1 \pmod{3}$，答案是 $\boxed{\textbf{(C) }30}$。

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