2023 AMC 12A Problems/Problem 4: Difference between revisions

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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 $8^5 \cdot 5^{10} \cdot 15^5$ ?

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

Solution 1

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

$10^{15}$ has $16$ 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, IT IS 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 (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

Video Solution Fast and Easy by DR.GOOGLE

https://www.youtube.com/watch?v=Q6-MiP6ZTYs

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

@@ Line 4: / Line 4: @@
 How many digits are in the base-ten representation of <math>8^5 \cdot 5^{10} \cdot 15^5</math>?
-<math>\textbf{(A)}~14\qquad\textbf{(B)}~15\qquad\textbf{(C)}~16\qquad\textbf{(D)}~17\qquad\textbf{(E)}~18\qquad</math>
+<math>\textbf{(A) } 14 \qquad\textbf{(B) }15 \qquad\textbf{(C) }16 \qquad\textbf{(D) }17 \qquad\textbf{(E) } 18</math>
 ==Solution 1==
-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{\textbf{(E) }18}}</math>
+Prime factorizing this gives us <math>2^{15}\cdot3^{5}\cdot5^{15}=10^{15}\cdot3^5=243\cdot10^{15}</math>.
+<math>10^{15}</math> has <math>16</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>
+<math>2.43*10^{17}</math> has <math>18</math> digits
 ~zhenghua
+==Solution 2 (Only if you don't know how to do the rest of the problems and have about 20 minutes left, IT IS  recommended)==
+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>
+~andliu766
+==Solution 3 (Similar to Solution 1)==
+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>
+~leon_0iler
+==Video Solution by Little Fermat==
+https://youtu.be/h2Pf2hvF1wE?si=mJjq7vPLptdSe0AJ&t=872
+~little-fermat
+==Video Solution (easy to digest) by Power Solve==
+https://youtu.be/Od1Spf3TDBs
-==Video Solution by Math-X (First understand the problem!!!)==
+== Video Solution by CosineMethod [🔥Fast and Easy🔥]==
-https://youtu.be/cMgngeSmFCY?si=b1khjbMn1i5rApCe&t=903
+https://www.youtube.com/watch?v=laHiorWO1zo
 ==Video Solution==
@@ Line 20: / Line 40: @@
 ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
+==Video Solution (⚡ Under 2 min ⚡)==
+https://youtu.be/Xy8vyymlPBg
+~Education, the Study of Everything
+==Video Solution Fast and Easy by DR.GOOGLE==
+https://www.youtube.com/watch?v=Q6-MiP6ZTYs
 ==See Also==

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