Try this beautiful problem from Algebra based on LCM from AMC-8, 2016.

## Problem based on LCM – AMC 8, 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 $c$?

- \(26\)
- \(20\)
- \(28\)

**Key Concepts**

Algebra

Divisor

multiplication

## Check the Answer

But try the problem first…

Answer:20

AMC-8, 2016 problem 20

Challenges and Thrills of Pre College Mathematics

## Try with Hints

First hint

We have to find out the least common multiple of $a$ and $c$.if you know the value of \(a\) and \(c\) then you can easily find out the required LCM. Can you find out the value of \(a\) and \(c\)?

Can you now finish the problem ……….

Second Hint

Given that the least common multiple of $a$ and $b$ is $12$, and the least common multiple of $b$ and $c$ is $15$ .then b must divide 12 and 15. There is only one possibility that b=3 which divide 12 and 15. therefore \(a\)=\(\frac{12}{3}=4\)

can you finish the problem……..

Final Step

so\(b\)=3. Given that LCM of \(b\) and \(c\) is 15. Therefore c=5

Now lcm of \(a\) and \(c\) that is lcm of 4 and 5=20

## Other useful links

- https://www.cheenta.com/perfect-cubes-algebra-amc-8-2018-problem-25/
- https://www.youtube.com/watch?v=hlWgXiqTHh8

Google