Question

What is the least common multiple of 6, 11, and 18?

Answer 1

Hint: We are first going to define what LCM is and explain it with a small basic example. LCM basically means the smallest number which is a common multiple for all the terms. For this, we find out all the factors for the numbers whose LCM is to be found. We then take the common factors once and the remaining terms and write the LCM.

Complete step-by-step solution:
In order to solve this question, let us first explain the concept of LCM or least common multiple. Least common multiple means the smallest number which is a multiple for the given set of numbers. Let us consider an example of say 3 and 4. We are supposed to find the LCM of the two numbers. For this, we write down the factors of the two numbers as follows:
$\Rightarrow 3=3\times 1$
$\Rightarrow 4=2\times 2\times 1$
We then need to write down the numbers common to both once and write the remaining numbers as the same.
$\Rightarrow LCM=3\times 2\times 2\times 1=12$
Hence, the LCM of 3 and 4 is 12.
Similarly, we are required to find the LCM of the numbers given in question which are 6, 11 and 18. We shall now write down all the factors of the given numbers 6,11 and 18.
Factorising 6:
$\Rightarrow 6=3\times 2\times 1$
Factorising 11:
$\Rightarrow 11=11\times 1$
Factorising 18:
$\Rightarrow 18=3\times 3\times 2\times 1$
Taking 2 and 3 only once, which is common to 6 and 18, we then write all the other terms as the same in the form of a product.
$\Rightarrow LCM=11\times 3\times 3\times 2\times 1=198$
Hence, the LCM of 6, 11 and 18 is 198.

Note: We can also solve this question by using the prime factorization method as shown below,
$\begin{align}
  & \text{ }2\left| \!{\underline {\,
  6,11,18 \,}} \right. \\
 & \text{ }3\left| \!{\underline {\,
  3,11,9 \,}} \right. \\
 & \text{ }3\left| \!{\underline {\,
  1,11,3 \,}} \right. \\
 & 11\left| \!{\underline {\,
  1,11,1 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1,1,1 \,}} \right. \\
\end{align}$
Taking a product of all these prime factors, we obtain the LCM which is the same as the value obtained in the solution, 198.