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Difference between revisions of "2024 AMC 12B Problems/Problem 6"

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==Problem 6==
 
==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)
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The national debt of the United States is on track to reach <math>5\times10^{13}</math> dollars by <math>2033</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) } 22 \qquad\textbf{(D) } 24 \qquad\textbf{(E) } 26</math>
 
<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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==Solution 1==
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Generally, number of digits of a number <math>n</math> in base <math>b</math>:
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The number of digits is just <math>\lceil \log_{5} 5\times 10^{13} \rceil</math>.
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<cmath>
 +
\text{Number of digits} = \lfloor \log_b n \rfloor + 1
 +
</cmath>
 +
In this question, it is <math>\lfloor \log_{5} 5\times 10^{13}\rfloor+1</math>.
 
Note that  
 
Note that  
 
<cmath>\log_{5} 5\times 10^{13}=1+\frac{13}{\log_{10} 5}</cmath>
 
<cmath>\log_{5} 5\times 10^{13}=1+\frac{13}{\log_{10} 5}</cmath>
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Hence, our answer is <math>\fbox{\textbf{(B) } 20}</math>
 
Hence, our answer is <math>\fbox{\textbf{(B) } 20}</math>
  
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~tsun26
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~tsun26 (small modification by notknowanything)
  
 
==Solution 2==
 
==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>.
 
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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==Solution 3==
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~sidkris
 +
 
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Note - Base Conversion Step
 +
 
 +
To convert the number <math>8192</math> from base 10 to base 5, we follow these steps:
 +
 
 +
1. Divide the number by 5 repeatedly, noting the quotient and remainder each time.
  
 +
2. Stop when the quotient becomes 0, then read the remainders from bottom to top.
 +
 +
<cmath>
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8192 \div 5 = 1638 \text{ remainder } 2
 +
</cmath>
 +
<cmath>
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1638 \div 5 = 327 \text{ remainder } 3
 +
</cmath>
 
<cmath>
 
<cmath>
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5 \times 10^{13} = 5 \times (2^{13} \times 5^{13}) = 2^{13} \times 5^{14}
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327 \div 5 = 65 \text{ remainder } 2
 
</cmath>
 
</cmath>
 
<cmath>
 
<cmath>
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2^{10} = 1024 \approx 10^3
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65 \div 5 = 13 \text{ remainder } 0
 
</cmath>
 
</cmath>
 
<cmath>
 
<cmath>
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2^{13} = 2^{10} \times 2^3 \approx 10^3 \times 8 = 8000
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13 \div 5 = 2 \text{ remainder } 3
 
</cmath>
 
</cmath>
 
<cmath>
 
<cmath>
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5 \times 10^{13} \approx 8000 \times 5^{14}
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2 \div 5 = 0 \text{ remainder } 2
 
</cmath>
 
</cmath>
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converted <math>8000</math> to base 5, divide <math>8000</math> repeatedly by 5 and keep track of the remainders:
 
  
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1. <math>8000 \div 5 = 1600</math>, remainder <math>0</math>
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Now, reading the remainders from bottom to top:<math> 2, 3, 0, 2, 3, 2 </math>.
  
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2. <math>1600 \div 5 = 320</math>, remainder <math>0</math>
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Thus, <math>8192</math> in base 5 is:
  
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3. <math>320 \div 5 = 64</math>, remainder <math>0</math>
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<cmath>
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\boxed{230232_5}
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4. <math>64 \div 5 = 12</math>, remainder <math>4</math>
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</cmath>
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~[https://artofproblemsolving.com/wiki/index.php/User:Cyantist luckuso]
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5. <math>12 \div 5 = 2</math>, remainder <math>2</math>
 
  
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6. <math>2 \div 5 = 0</math>, remainder <math>2</math>
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==Video Solution 1 by SpreadTheMathLove==
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https://www.youtube.com/watch?v=FUsMSwb-JUc
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Thus, <math>8000</math> in base 5 is <math>224000_5</math>, which has  6 digits
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==See also==
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When we multiply <math>224000_5</math> by <math>5^{14}</math>, the multiplication shifts the digits by 14 places to the left, adding 14 zeros.
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{{AMC12 box|year=2024|ab=B|num-b=5|num-a=7}}
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Thus, the total number of digits is:
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{{MAA Notice}}
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6 + 14 = <math>\fbox{\textbf{(B)} 20}</math>.
 
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~[https://artofproblemsolving.com/wiki/index.php/User:Cyantist luckuso]
 

Latest revision as of 14:54, 21 November 2024

Problem 6

The national debt of the United States is on track to reach $5\times10^{13}$ dollars by $2033$. How many digits does this number of dollars have when written as a numeral in base 5? (The approximation of $\log_{10} 5$ as $0.7$ is sufficient for this problem)

$\textbf{(A) } 18 \qquad\textbf{(B) } 20 \qquad\textbf{(C) } 22 \qquad\textbf{(D) } 24 \qquad\textbf{(E) } 26$

Solution 1

Generally, number of digits of a number $n$ in base $b$: \[\text{Number of digits} = \lfloor \log_b n \rfloor + 1\] In this question, it is $\lfloor \log_{5} 5\times 10^{13}\rfloor+1$. Note that \[\log_{5} 5\times 10^{13}=1+\frac{13}{\log_{10} 5}\] \[\approx 1+\frac{13}{0.7}\] \[\approx 19.5\]

Hence, our answer is $\fbox{\textbf{(B) } 20}$

~tsun26 (small modification by notknowanything)

Solution 2

We see that $5\times 10^{13} = 2^{13} \cdot 5^{14}$ and $2^{13} = 8192$. Converting this to base $5$ gives us $230232$ (trust me it doesn't take that long). So the final number in base $5$ is $230232$ with $14$ zeroes at the end, which gives us $6 + 14 = 20$ digits. So the answer is $\fbox{\textbf{(B)} 20}$.

~sidkris

Note - Base Conversion Step

To convert the number $8192$ from base 10 to base 5, we follow these steps:

1. Divide the number by 5 repeatedly, noting the quotient and remainder each time.

2. Stop when the quotient becomes 0, then read the remainders from bottom to top.

\[8192 \div 5 = 1638 \text{ remainder } 2\] \[1638 \div 5 = 327 \text{ remainder } 3\] \[327 \div 5 = 65 \text{ remainder } 2\] \[65 \div 5 = 13 \text{ remainder } 0\] \[13 \div 5 = 2 \text{ remainder } 3\] \[2 \div 5 = 0 \text{ remainder } 2\]

Now, reading the remainders from bottom to top:$2, 3, 0, 2, 3, 2$.

Thus, $8192$ in base 5 is:

\[\boxed{230232_5}\] ~luckuso

Video Solution 1 by SpreadTheMathLove

https://www.youtube.com/watch?v=FUsMSwb-JUc

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

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