AMC12 2024 B

AMC12 2024 B · Q6

AMC12 2024 B · Q6. It mainly tests Rounding & estimation, Base representation.

The national debt of the United States is on track to reach $5\times10^{13}$ dollars by $2033$. How many digits does this number of dollars have when written as a numeral in base $5$? (The approximation of $\log_{10} 5$ as $0.7$ is sufficient for this problem)

美国国债预计到2033年将达到$5\times10^{13}$美元。这个美元数额用5进制书写时有几位？（本题中近似$\log_{10} 5 \approx 0.7$即可）

(A) 18 18

(B) 20 20

(D) 24 24

(E) 26 26

Answer

Correct choice: (B)

正确答案：（B）

Solution

Generally, the number of digits of number $n$ in base $b$ is \[\lfloor \log_b n \rfloor + 1.\] In this question, it is $\lfloor \log_{5} (5\times 10^{13})\rfloor+1$. Note that \begin{align*} \log_{5} (5\times 10^{13}) &= 1+\frac{13}{\log_{10} 5} \\ &\approx 1+\frac{13}{0.7} \\ &\approx 19.5 \end{align*} Hence, our answer is $\fbox{\textbf{(B)} 20}$

一般地，数字$n$在$b$进制中的位数为 \[\lfloor \log_b n \rfloor + 1.\] 本题中为$\lfloor \log_{5} (5\times 10^{13})\rfloor+1$。注意到 \begin{align*} \log_{5} (5\times 10^{13}) &= 1+\frac{13}{\log_{10} 5} \\ &\approx 1+\frac{13}{0.7} \\ &\approx 19.5 \end{align*} 因此，答案为$\fbox{\textbf{(B)} 20}$

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