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Difference between revisions of "2013 AMC 12B Problems/Problem 9"

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==Problem==
 
==Problem==
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What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides <math>12!</math> ?
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What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides <math>12!</math>?
  
 
<math>\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12 </math>
 
<math>\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12 </math>
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==Solution==
 
==Solution==
  
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Looking at the prime numbers under 12, we see that there are <math>\lfloor\frac{12}{2}\rfloor+\lfloor\frac{12}{2^2}\rfloor+\lfloor\frac{12}{2^3}\rfloor=6+3+1=10</math> factors of 2, <math>\lfloor\frac{12}{3}\rfloor+\lfloor\frac{12}{3^2}\rfloor=4+1=5</math> factors of 3, and <math>\lfloor\frac{12}{5}\rfloor=2</math> factors of 5. All greater primes are represented once or not at all in <math>12!</math>, so they cannot be part of the square. Since we are looking for a perfect square, the exponents on its prime factors must be even, so we can only use <math>4</math> of the <math>5</math> factors of <math>3</math>. The prime factorization of the square is therefore <math>2^{10}*3^4*5^2</math>. To find the square root of this, we halve the exponents, leaving <math>2^5*3^2*5</math>. The sum of the exponents is <math>\boxed{\textbf{(C) }8}</math>
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Looking at the prime numbers under <math>12</math>, we see that there are <math>\left\lfloor\frac{12}{2}\right\rfloor+\left\lfloor\frac{12}{2^2}\right\rfloor+\left\lfloor\frac{12}{2^3}\right\rfloor=6+3+1=10</math> factors of <math>2</math>, <math>\left\lfloor\frac{12}{3}\right\rfloor+\left\lfloor\frac{12}{3^2}\right\rfloor=4+1=5</math> factors of <math>3</math>, and <math>\left\lfloor\frac{12}{5}\right\rfloor=2</math> factors of <math>5</math>. All greater primes are represented once or none in <math>12!</math>, so they cannot be part of the square. Since we are looking for a perfect square, the exponents on its prime factors must be even, so we can only use <math>4</math> of the <math>5</math> factors of <math>3</math>. Therefore, the prime factorization of the square is <math>2^{10}\cdot3^4\cdot5^2</math>. To find the square root of this, we halve the exponents, leaving <math>2^5\cdot3^2\cdot5</math>. The sum of the exponents is <math>\boxed{\textbf{(C) }8}</math>
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== Video Solution by OmegaLearn==
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https://youtu.be/ZhAZ1oPe5Ds?t=2694
 +
 
 +
~ pi_is_3.14
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==Video Solution==
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https://youtu.be/a-3CAo4CoWc
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 +
~someone
  
 
== See also ==
 
== See also ==
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[[Category:Introductory Number Theory Problems]]
 
[[Category:Introductory Number Theory Problems]]
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{{MAA Notice}}

Latest revision as of 19:25, 7 August 2023

Problem

What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $12!$?

$\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12$

Solution

Looking at the prime numbers under $12$, we see that there are $\left\lfloor\frac{12}{2}\right\rfloor+\left\lfloor\frac{12}{2^2}\right\rfloor+\left\lfloor\frac{12}{2^3}\right\rfloor=6+3+1=10$ factors of $2$, $\left\lfloor\frac{12}{3}\right\rfloor+\left\lfloor\frac{12}{3^2}\right\rfloor=4+1=5$ factors of $3$, and $\left\lfloor\frac{12}{5}\right\rfloor=2$ factors of $5$. All greater primes are represented once or none in $12!$, so they cannot be part of the square. Since we are looking for a perfect square, the exponents on its prime factors must be even, so we can only use $4$ of the $5$ factors of $3$. Therefore, the prime factorization of the square is $2^{10}\cdot3^4\cdot5^2$. To find the square root of this, we halve the exponents, leaving $2^5\cdot3^2\cdot5$. The sum of the exponents is $\boxed{\textbf{(C) }8}$

Video Solution by OmegaLearn

https://youtu.be/ZhAZ1oPe5Ds?t=2694

~ pi_is_3.14

Video Solution

https://youtu.be/a-3CAo4CoWc

~someone

See also

2013 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
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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