2024 AMC 12B Problems/Problem 6

Problem 6

The national debt of the United States is on track to reach $5\times10^{13}$ dollars by $2023$ . 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)

$\textbf{(A) } 18 \qquad\textbf{(B) } 20 \qquad\textbf{(C) } 22 \qquad\textbf{(D) } 24 \qquad\textbf{(E) } 26$

Solution

The number of digits is just $\lceil \log_{5} 5\times 10^{13} \rceil$ . Note that $\log_{5} 5\times 10^{13}=1+\frac{13}{\log_{10} 5}$ $\approx 1+\frac{13}{0.7}$ $\approx 19.5$

Hence, our answer is $\fbox{\textbf{(B) } 20}$

~tsun26

Solution 2

We see that $5\times 10^{13} = 2^{13} \cdot 5^{14}$ and $2^{13} = 8192$ . Converting this to base $5$ gives us $230232$ (trust me it doesn't take that long). So the final number in base $5$ is $230232$ with $14$ zeroes at the end, which gives us $6 + 14 = 20$ digits. So the answer is $\fbox{\textbf{(B)} 20}$ .

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2024 AMC 12B Problems/Problem 6

Problem 6

Solution

Solution 2