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Difference between revisions of "2024 AMC 12B Problems/Problem 6"

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~sidkris
 
~sidkris
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==Solution 3==
 
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<cmath>
 
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5 \times 10^{13} = 5 \times (2^{13} \times 5^{13}) = 2^{13} \times 5^{14}
 
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</cmath>
 
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<cmath>
 
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2^{10} = 1024 \approx 10^3
 
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</cmath>
 
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<cmath>
 
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2^{13} = 2^{10} \times 2^3 \approx 10^3 \times 8 = 8000
 
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</cmath>
 
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<cmath>
 
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5 \times 10^{13} \approx 8000 \times 5^{14}
 
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</cmath>
 
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converted <math>8000</math> to base 5, divide <math>8000</math> repeatedly by 5 and keep track of the remainders:
 
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1. <math>8000 \div 5 = 1600</math>, remainder <math>0</math>
 
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2. <math>1600 \div 5 = 320</math>, remainder <math>0</math>
 
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3. <math>320 \div 5 = 64</math>, remainder <math>0</math>
 
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4. <math>64 \div 5 = 12</math>, remainder <math>4</math>
 
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5. <math>12 \div 5 = 2</math>, remainder <math>2</math>
 
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6. <math>2 \div 5 = 0</math>, remainder <math>2</math>
 
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Thus, <math>8000</math> in base 5 is <math>224000_5</math>, which has  6 digits
 
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When we multiply <math>224000_5</math> by <math>5^{14}</math>, the multiplication shifts the digits by 14 places to the left, adding 14 zeros.
 
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Thus, the total number of digits is:
 
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6 + 14 = <math>\fbox{\textbf{(B)} 20}</math>.
 
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~[https://artofproblemsolving.com/wiki/index.php/User:Cyantist luckuso]
 

Revision as of 02:38, 14 November 2024

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

~sidkris

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