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Difference between revisions of "2023 AMC 12A Problems/Problem 4"

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{{duplicate|[[2023 AMC 10A Problems/Problem 5|2023 AMC 10A #5]] and [[2023 AMC 12A Problems/Problem 4|2023 AMC 12A #4]]}}
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==Problem==
 
==Problem==
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How many digits are in the base-teen representation of <math>8^5 \cdot 5^{10} \cdot 15^5</math>?
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How many digits are in the base-ten representation of <math>8^5 \cdot 5^{10} \cdot 15^5</math>?
  
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<cmath>\textbf{(A)}~14\qquad\textbf{(B)}~15\qquad\textbf{(C)}~16\qquad\textbf{(D)}~17\qquad\textbf{(E)}~18\qquad</cmath>
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<math>\textbf{(A)}~14\qquad\textbf{(B)}~15\qquad\textbf{(C)}~16\qquad\textbf{(D)}~17\qquad\textbf{(E)}~18\qquad</math>
  
 
==Solution 1==
 
==Solution 1==
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Prime factorizing this gives us <math>2^{15}\cdot3^{5}\cdot5^{15}=10^{15}\cdot3^5=243\cdot10^{15}</math>. <math>10^{15}</math> gives us <math>15</math> digits, and <math>243</math> gives us <math>3</math> digits. <math>15+3=\text{\boxed{(E) 18}}</math>
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Prime factorizing this gives us <math>2^{15}\cdot3^{5}\cdot5^{15}=10^{15}\cdot3^5=243\cdot10^{15}</math>.
 +
 
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<math>10^{15}</math> has <math>15</math> digits and <math>243</math> = <math>2.43*10^{2}</math> gives us <math>3</math> more digits. <math>16+2=\text{\boxed{\textbf{(E) }18}}</math>
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<math>2.43*10^{17}</math> has <math>18</math> digits
  
 
~zhenghua
 
~zhenghua
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==Solution 2 (Only if you don't know how to do the rest of the problems and have about 20 minutes left, not recommended)==
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Multiplying it out, we get that <math>8^5 \cdot 5^{10} \cdot 15^5 = 243000000000000000</math>. Counting, we have the answer is <math>\text{\boxed{\textbf{(E) }18}}</math>
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~andliu766
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 +
==Solution 3 (Similar to Solution 1)==
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All the exponents have a common factor of <math>5</math> which we can factor out. This leaves us with <math>(8 \cdot 5^2 \cdot 15)^5 = (3000)^5 = (3 \cdot 1000)^5</math>. We can then distribute the power leaving us with <math>3^5 \cdot 10^{3 \cdot 5} = 243 \cdot 10^{15}</math>. This would be <math>243</math> followed by <math>15</math> zeros resulting in our answer being <math>15+3=\text{\boxed{\textbf{(E)}18}}</math>
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 +
~leon_0iler
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 +
==Video Solution by Little Fermat==
 +
https://youtu.be/h2Pf2hvF1wE?si=mJjq7vPLptdSe0AJ&t=872
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~little-fermat
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 +
==Video Solution by Math-X (First understand the problem!!!)==
 +
https://youtu.be/GP-DYudh5qU?si=1RDs-j8Cedw02bID&t=983
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 +
==Video Solution (easy to digest) by Power Solve==
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https://youtu.be/Od1Spf3TDBs
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 +
== Video Solution by CosineMethod [🔥Fast and Easy🔥]==
 +
 +
https://www.youtube.com/watch?v=laHiorWO1zo
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 +
==Video Solution==
 +
 +
https://youtu.be/MFPSxqtguQo
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 +
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
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 +
==Video Solution (⚡ Under 2 min ⚡)==
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https://youtu.be/Xy8vyymlPBg
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 +
~Education, the Study of Everything
  
 
==See Also==
 
==See Also==

Latest revision as of 06:21, 5 November 2024

The following problem is from both the 2023 AMC 10A #5 and 2023 AMC 12A #4, so both problems redirect to this page.

Problem

How many digits are in the base-ten representation of $8^5 \cdot 5^{10} \cdot 15^5$?

$\textbf{(A)}~14\qquad\textbf{(B)}~15\qquad\textbf{(C)}~16\qquad\textbf{(D)}~17\qquad\textbf{(E)}~18\qquad$

Solution 1

Prime factorizing this gives us $2^{15}\cdot3^{5}\cdot5^{15}=10^{15}\cdot3^5=243\cdot10^{15}$.

$10^{15}$ has $15$ digits and $243$ = $2.43*10^{2}$ gives us $3$ more digits. $16+2=\text{\boxed{\textbf{(E) }18}}$

$2.43*10^{17}$ has $18$ digits

~zhenghua

Solution 2 (Only if you don't know how to do the rest of the problems and have about 20 minutes left, not recommended)

Multiplying it out, we get that $8^5 \cdot 5^{10} \cdot 15^5 = 243000000000000000$. Counting, we have the answer is $\text{\boxed{\textbf{(E) }18}}$ ~andliu766

Solution 3 (Similar to Solution 1)

All the exponents have a common factor of $5$ which we can factor out. This leaves us with $(8 \cdot 5^2 \cdot 15)^5 = (3000)^5 = (3 \cdot 1000)^5$. We can then distribute the power leaving us with $3^5 \cdot 10^{3 \cdot 5} = 243 \cdot 10^{15}$. This would be $243$ followed by $15$ zeros resulting in our answer being $15+3=\text{\boxed{\textbf{(E)}18}}$

~leon_0iler

Video Solution by Little Fermat

https://youtu.be/h2Pf2hvF1wE?si=mJjq7vPLptdSe0AJ&t=872 ~little-fermat

Video Solution by Math-X (First understand the problem!!!)

https://youtu.be/GP-DYudh5qU?si=1RDs-j8Cedw02bID&t=983

Video Solution (easy to digest) by Power Solve

https://youtu.be/Od1Spf3TDBs

Video Solution by CosineMethod [🔥Fast and Easy🔥]

https://www.youtube.com/watch?v=laHiorWO1zo

Video Solution

https://youtu.be/MFPSxqtguQo

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Video Solution (⚡ Under 2 min ⚡)

https://youtu.be/Xy8vyymlPBg

~Education, the Study of Everything

See Also

2023 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
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 10 Problems and Solutions
2023 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
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. AMC logo.png

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