Difference between revisions of "2024 AMC 12B Problems/Problem 6"
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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) | 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) | ||
| − | <math>\textbf{(A) } 18 \qquad\textbf{(B) } 20 \qquad\textbf{(C) } | + | <math>\textbf{(A) } 18 \qquad\textbf{(B) } 20 \qquad\textbf{(C) } 22 \qquad\textbf{(D) } 24 \qquad\textbf{(E) } 26</math> |
| + | |||
| + | ==Solution== | ||
| + | |||
| + | The number of digits is just <math>\lceil \log_{5} 5\times 10^{13} \rceil</math>. | ||
| + | Note that | ||
| + | <cmath>\log_{5} 5\times 10^{13}=1+\frac{13}{\log_{10} 5}</cmath> | ||
| + | <cmath>\approx 1+\frac{13}{0.7}</cmath> | ||
| + | <cmath>\approx 19.5</cmath> | ||
| + | |||
| + | Hence, our answer is <math>\textbf{(B) } 20</math> | ||
Revision as of 01:19, 14 November 2024
Problem 6
The national debt of the United States is on track to reach 
dollars by 
. How many digits does this number of dollars have when written as a numeral in base 5? (The approximation of 
as 
is sufficient for this problem)
Solution
The number of digits is just 
.
Note that
![\[\log_{5} 5\times 10^{13}=1+\frac{13}{\log_{10} 5}\]](http://latex.artofproblemsolving.com/d/1/e/d1e4245753973d10ebc097221da7f3bcb66a510a.png)
![\[\approx 1+\frac{13}{0.7}\]](http://latex.artofproblemsolving.com/c/b/6/cb6c371178c546b7b071a74c8a98d5852f2f520b.png)
![\[\approx 19.5\]](http://latex.artofproblemsolving.com/1/7/b/17bce86bb35886da1b4460f7e79c804aeefddec5.png)
Hence, our answer is 
