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

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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==
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>, algebraically, it's some multiple of <math>b</math> and from looking at the numbers, we are sure that it is 3, thus <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{A}</math>
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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>, algebraically, it's some multiple of <math>b</math> and from looking at the numbers, we are sure that it is 3, thus <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 ==
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https://youtu.be/HISL2-N5NVg?t=2340
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~ pi_is_3.14
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==See Also==
 
{{AMC8 box|year=2016|num-b=19|num-a=21}}
 
{{AMC8 box|year=2016|num-b=19|num-a=21}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 17:33, 21 April 2021

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 $15$, algebraically, it's some multiple of $b$ and from looking at the numbers, we are sure that it is 3, thus $b$ 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

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

~ pi_is_3.14


See Also

2016 AMC 8 (ProblemsAnswer KeyResources)
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

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