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Difference between revisions of "2013 AMC 8 Problems/Problem 10"

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==Problem==
 
==Problem==
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What is the ratio of the least comon multiple of 180 and 594 to the greatest comon factor of 180 and 594?
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What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?
  
 
<math>\textbf{(A)}\ 110 \qquad \textbf{(B)}\ 165 \qquad \textbf{(C)}\ 330 \qquad \textbf{(D)}\ 625 \qquad \textbf{(E)}\ 660</math>
 
<math>\textbf{(A)}\ 110 \qquad \textbf{(B)}\ 165 \qquad \textbf{(C)}\ 330 \qquad \textbf{(D)}\ 625 \qquad \textbf{(E)}\ 660</math>
  
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==Solution 1==
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==Solution 1==  
 
To find either the LCM or the GCF of two numbers, always prime factorize first.
 
To find either the LCM or the GCF of two numbers, always prime factorize first.
  

Latest revision as of 21:42, 5 January 2024

Problem

What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?

$\textbf{(A)}\ 110 \qquad \textbf{(B)}\ 165 \qquad \textbf{(C)}\ 330 \qquad \textbf{(D)}\ 625 \qquad \textbf{(E)}\ 660$

Solution 1

To find either the LCM or the GCF of two numbers, always prime factorize first.

The prime factorization of $180 = 3^2 \times 5 \times 2^2$.

The prime factorization of $594 = 3^3 \times 11 \times 2$.

Then, to find the LCM, we have to find the greatest power of all the numbers there are; if one number is one but not the other, use it (this is $3^3, 5, 11, 2^2$). Multiply all of these to get 5940.

For the GCF of 180 and 594, use the least power of all of the numbers that are in both factorizations and multiply. $3^2 \times 2$ = 18.

Thus the answer = $\frac{5940}{18}$ = $\boxed{\textbf{(C)}\ 330}$.

Solution 2

We start off with a similar approach as the original solution. From the prime factorizations, the GCF is $18$.

It is a well known fact that $\gcd(m,n)\times \operatorname{lcm}(m,n)=|mn|$. So we have, $18\times \operatorname{lcm} (180,594)=594\times 180$.

Dividing by $18$ yields $\operatorname{lcm} (180,594)=594\times 10=5940$.

Therefore, $\frac{\operatorname{lcm} (180,594)}{\operatorname{gcf}(180,594)}=\frac{5940}{18}=\boxed{\textbf{(C)}\ 330}$.

Solution 3

From Solution 1, the prime factorization of $180 = 2^2 \cdot 3^2 \cdot 5$. The prime factorization of $594 = 2 \cdot 3^3 \cdot 11$. Hence, $\operatorname{lcm} (180,594) = 2^2 \cdot 3^3 \cdot 5 \cdot 11$, and $\operatorname{gcf} (180,594) = 2 \cdot 3^2$. Therefore, $\frac{\operatorname{lcm} (180,594)}{\operatorname{gcf} (180,594)} = \frac{2^2 \cdot 3^3 \cdot 5 \cdot 11}{2 \cdot 3^2} = 2 \cdot 3 \cdot 5 \cdot 11 = 330 \Longrightarrow \boxed{\textbf{(C)}\ 330}$

Video Solution by OmegaLearn

https://youtu.be/CWZkTCNu42o?t=31

~ pi_is_3.14

Video Solution

https://youtu.be/ZtjQTa2hWmw ~savannahsolver

See Also

2013 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
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 AJHSME/AMC 8 Problems and Solutions

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