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

Latest revision as of 16:22, 7 September 2025

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

Solution 2

The factors of 12 are 12,6,4,3,2,1. The factors of 15 are 1,3,5,15. The 2 numbers that repeat are 1 and 3 so b either has to be 1 or 3. If b is 3 then a is 4 and c is 5 and the least common multiple of 4 and 5 are 20. We don't have to test 1 because 20 is the lowest answer, so if b equaling 1 resulted in the least common multiple being less that 20 then the correct answer won't be there. So the answer is $\boxed{\textbf{(A)} 20}$ .

Solution 3

Let's say that $a = 4, b = 3, c = 5$ , this works so clearly $GCF(4, 5) = \boxed{\textbf{(A)} 20}$ . - JasonDaGoat

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

@@ Line 11: / Line 11: @@
 The factors of 12 are 12,6,4,3,2,1. The factors of 15 are 1,3,5,15. The 2 numbers that repeat are 1 and 3 so b either has to be 1 or 3. If b is 3 then a is 4 and c is 5 and the least common multiple of 4 and 5 are 20. We don't have to test 1 because 20 is the lowest answer, so if b equaling 1 resulted in the least common multiple being less that 20 then the correct answer won't be there. So the answer is <math>\boxed{\textbf{(A)} 20}</math>.
 ===Solution 3===
-Let's say that <math>a = 4, b = 3, c = 5</math>, this works so clearly <math>GCF(4, 5) = \boxed{\textbf{(A)} 20}</math>.
+Let's say that <math>a = 4, b = 3, c = 5</math>, this works so clearly <math>GCF(4, 5) = \boxed{\textbf{(A)} 20}</math>. - JasonDaGoat
 ==Video Solution by Pi Academy==

2016 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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