Difference between revisions of "2024 AMC 12B Problems/Problem 6"
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Revision as of 09:58, 14 November 2024
Contents
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
See also
| 2024 AMC 12B (Problems • Answer Key • Resources) | |
| 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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
