Difference between revisions of "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 $20 \rightarrow \boxed{\textbf{(A)} 20}$

~ pi_is_3.14