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 1

Generally, number of digits of a number $n$ in base $b$: \[\text{Number of digits} = \lfloor \log_b n \rfloor + 1\] In this question, it is $\lfloor \log_{5} 5\times 10^{13}\rfloor+1$. 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 (small modification by notknowanything)

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 $\fbox{\textbf{(B)} 20}$.

~sidkris

Note - Base Conversion Step

To convert the number $8192$ from base 10 to base 5, we follow these steps:

1. Divide the number by 5 repeatedly, noting the quotient and remainder each time.

2. Stop when the quotient becomes 0, then read the remainders from bottom to top.

\[8192 \div 5 = 1638 \text{ remainder } 2\] \[1638 \div 5 = 327 \text{ remainder } 3\] \[327 \div 5 = 65 \text{ remainder } 2\] \[65 \div 5 = 13 \text{ remainder } 0\] \[13 \div 5 = 2 \text{ remainder } 3\] \[2 \div 5 = 0 \text{ remainder } 2\]

Now, reading the remainders from bottom to top:$2, 3, 0, 2, 3, 2$.

Thus, $8192$ in base 5 is:

\[\boxed{230232_5}\] ~luckuso

Video Solution 1 by SpreadTheMathLove

https://www.youtube.com/watch?v=FUsMSwb-JUc

See also

2024 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 5
Followed by
Problem 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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