2016 AMC 8 Problems/Problem 20

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 (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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