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| − | ==Problem 6==
| + | #redirect[[2024 AMC 10B Problems/Problem 15]] |
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| − | The national debt of the United States is on track to reach <math>5\times10^{13}</math> dollars by <math>2023</math>. How many digits does this number of dollars have when written as a numeral in base 5? (The approximation of <math>\log_{10} 5</math> as <math>0.7</math> is sufficient for this problem)
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| − | <math>\textbf{(A) } 18 \qquad\textbf{(B) } 20 \qquad\textbf{(C) } 22 \qquad\textbf{(D) } 24 \qquad\textbf{(E) } 26</math>
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| − | ==Solution==
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| − | The number of digits is just <math>\lceil \log_{5} 5\times 10^{13} \rceil</math>.
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| − | Note that
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| − | <cmath>\log_{5} 5\times 10^{13}=1+\frac{13}{\log_{10} 5}</cmath>
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| − | <cmath>\approx 1+\frac{13}{0.7}</cmath>
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| − | <cmath>\approx 19.5</cmath>
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| − | Hence, our answer is <math>\fbox{\textbf{(B) } 20}</math>
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| − | ~tsun26
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