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Difference between revisions of "2016 AMC 8 Problems/Problem 20"

(Created page with "The least common multiple of <math>a</math> and <math>b</math> is <math>12</math>, and the least common multiple of <math>b</math> and <math>c</math> is <math>15</math>. What...")
 
 
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==Problem==
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The least common multiple of <math>a</math> and <math>b</math> is <math>12</math>, and the least common multiple of <math>b</math> and <math>c</math> is <math>15</math>. What is the least possible value of the least common multiple of <math>a</math> and <math>c</math>?
 
The least common multiple of <math>a</math> and <math>b</math> is <math>12</math>, and the least common multiple of <math>b</math> and <math>c</math> is <math>15</math>. What is the least possible value of the least common multiple of <math>a</math> and <math>c</math>?
  
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==Solution==
 
==Solution==
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We wish to find possible values of <math>a</math>, <math>b</math>, and <math>c</math>. By finding the greatest common factor of <math>12</math> and <math>15</math>, we can find that <math>b</math> is 3. Moving on to <math>a</math> and <math>c</math>, in order to minimize them, we wish to find the least such that the least common multiple of <math>a</math> and <math>3</math> is <math>12</math>, <math>\rightarrow 4</math>. Similarly, with <math>3</math> and <math>c</math>, we obtain <math>5</math>. The least common multiple of <math>4</math> and <math>5</math> is <math>20 \rightarrow \boxed{\textbf{(A)} 20}</math>
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==Video Solution by Pi Academy==
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https://youtu.be/wvRmxjwOUHY?si=mNtAIGDHVdPaKWUX
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 +
 +
==Video Solution(CREATIVE THINKING + ANALYSIS!!!)==
 +
https://youtu.be/_-xC-qQMCbk
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 +
~Education, the Study of Everything
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 +
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== Video Solution by OmegaLearn==
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https://youtu.be/HISL2-N5NVg?t=2340
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~ pi_is_3.14
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== Video Solution ==
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https://youtu.be/7tGFq07njVo
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~savannahsolver
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==See Also==
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{{AMC8 box|year=2016|num-b=19|num-a=21}}
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{{MAA Notice}}

Latest revision as of 10:00, 9 November 2024

Problem

The least common multiple of $a$ and $b$ is $12$, and the least common multiple of $b$ and $c$ is $15$. What is the least possible value of the least common multiple of $a$ and $c$?

$\textbf{(A) }20\qquad\textbf{(B) }30\qquad\textbf{(C) }60\qquad\textbf{(D) }120\qquad \textbf{(E) }180$

Solution

We wish to find possible values of $a$, $b$, and $c$. By finding the greatest common factor of $12$ and $15$, we can find that $b$ is 3. Moving on to $a$ and $c$, in order to minimize them, we wish to find the least such that the least common multiple of $a$ and $3$ is $12$, $\rightarrow 4$. Similarly, with $3$ and $c$, we obtain $5$. The least common multiple of $4$ and $5$ is $20 \rightarrow \boxed{\textbf{(A)} 20}$

Video Solution by Pi Academy

https://youtu.be/wvRmxjwOUHY?si=mNtAIGDHVdPaKWUX


Video Solution(CREATIVE THINKING + ANALYSIS!!!)

https://youtu.be/_-xC-qQMCbk

~Education, the Study of Everything


Video Solution by OmegaLearn

https://youtu.be/HISL2-N5NVg?t=2340

~ pi_is_3.14

Video Solution

https://youtu.be/7tGFq07njVo

~savannahsolver

See Also

2016 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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