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

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suppose that a and b is 12 and 3, and b and c is 3 and 15. then the lcm of 12 and 15 is 60.. aka c.
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solution:
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suppose that a and b is 12 and 3, and b and c is 3 and 15. then the lcm of 12 and 15 is 60. aka c.

Revision as of 19:44, 23 November 2016

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 $20 \rightarrow \boxed{A}$

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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solution: suppose that a and b is 12 and 3, and b and c is 3 and 15. then the lcm of 12 and 15 is 60. aka c.

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