Difference between revisions of "2024 AMC 12B Problems/Problem 6"
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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>. | + | 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>. |
~sidkris | ~sidkris |
Latest revision as of 02:38, 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
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