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 $15$ , 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 $20 \rightarrow \boxed{\textbf{(A)} 20}$

2016 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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All AJHSME/AMC 8 Problems and Solutions

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2016 AMC 8 Problems/Problem 20

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Solution