Question

How do you find the least common multiple of 10, 4, 18?

Answer 1

Hint: We are given three number 10, 4, 18, we are asked to find the least common multiple (LCM) of them, to do so we will learn about LCM then we will start with basic example once we get the grip then we will find the lowest common multiple of 10, 4, 18, we will factor 10, 4, 18, into its prime multiple then we try to make pair of same term and the remaining term.

Complete step by step solution:
We are given 3 terms as 10, 4, and 18.
We have to find lowest common multiple; we will learn what does least common multiple signifies.
Least common multiple as the name suggests we have to look for those values which is a common multiple of all the given terms, along with being common multiple it should also be the least one of all the multiple.
Let us work on an example, consider we have 2 and 3 so we can see that 6 is the multiple of 2 and 3, 12, 18,24 are also multiple of 2 and 3 but out of 6,12,18,24 etc the least one is 6, so least common multiple is 6.
Least common multiple is also denoted as LCM.
Now to find the least common multiple, we will prime factor all the terms and write the different possible factors.
For example, say we have to find LCMs of 2 and 4. So we are prime factor of 2 and 4.
So, $2=2\times 1$
$4=2\times 2\times 1$
Now to find the lowest common multiple, we write common things to both just 1 time and rest remaining as it is.
In $2=2\times 1\text{ and }4=2\times 2\times 1$ .
‘2’ is common and then we leave with 2 and 1.
So, LCM is $2\times 2\times 1$ which is ‘4’.
Now, we work on our problem.
We have 10, 4, and 18.
So, we factor each term.
We have
 $\begin{align}
  & 10=2\times 5\times 1 \\
 & 4=2\times 2\times 1 \\
 & 18=2\times 3\times 3\times 1 \\
\end{align}$
So, we can take ‘2’ as common to all. We will write the common ‘2’ just once.
Now we left with ‘5’ in 10, ‘2’ in 4 and $3\times 3$ in 18.
So, we write them as well.
So, the lowest common multiple is $LCM\left( 10,4,18 \right)=2\times 5\times 2\times 3\times 3$ .
We product and get –
LCM=180.

Note: We can also find the lowest common multiple by the common prime factorization, in this the term which is common to all separate out at once.
Now we do find the lowest common multiple by common prime factorization method.
$\begin{align}
  & 2\left| \!{\underline {\,
  10,4,18 \,}} \right. \\
 & 2\left| \!{\underline {\,
  5,2,9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  5,1,9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  5,1,3 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5,1,1 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1,1,1 \,}} \right. \\
\end{align}$
So, the lowest common multiple of 10, 4, and 18 is $LCM\left( 10,4,18 \right)=2\times 2\times 3\times 3\times 5=180$ .