Difference between revisions of "2024 AMC 12B Problems/Problem 6"
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| + | ==Solution 2== | ||
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| + | We see that <math>5\times 10^{13} = 2^{13} \cdot 5^{14}</math> and <math>2^{13} = 8192</math>. Converting this to base <math>5</math> gives us <math>230232</math> (trust me it doesn't take that long). So the final number in base <math>5</math> is <math>230232</math> with <math>14</math> zeroes at the end, which gives us <math>6 + 14 = 20</math> digits. So the answer is <math>\fbox{\textbf{(B) } 20}</math>. | ||
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| + | ~sidkris | ||
Revision as of 02:13, 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 
~tsun26
Solution 2
We see that 
and 
. Converting this to base 
gives us 
(trust me it doesn't take that long). So the final number in base is with 
zeroes at the end, which gives us 
digits. So the answer is .
~sidkris
