Question

What is the Least Common Multiple of 6, 8 and 9?

Answer 1

Hint: To calculate the LCM of three numbers, we should first find the prime factorisation of all three numbers, and then, we can write LCM by multiplying the prime factors as many times as the maximum number of times of their occurrence in any given number.

Complete step by step answer:
Least Common Multiple or LCM of three numbers, a, b and c, is defined as the smallest number possible that is perfectly divisible by all three a, b and c.
To use the prime factorisation method, we will first have to calculate the prime factorisation, and then list the prime factors as many times as the maximum number of times of their occurrence.
Here in this question, we need to calculate the LCM of 6, 8 and 9.
The prime factorisation of 6 is:
\[\begin{align}
  & 2\left| \!{\underline {\,
  6 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
$\therefore 6=2\times 3$
The prime factorisation of 8 is:
\[\begin{align}
  & 2\left| \!{\underline {\,
  8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
$\therefore 8=2\times 2\times 2$
 The prime factorisation of 9 is:
\[\begin{align}
  & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
$\therefore 9=3\times 3$
Now, for the prime factor 2, we have,
The number of times of occurrence of 2 in prime factorisation of 6 = 1
The number of times of occurrence of 2 in prime factorisation of 8 = 3
The number of times of occurrence of 2 in prime factorisation of 9 = 0
So, the maximum number of times of occurrence of 2 = 3 …(i)
Similarly, for the prime factor 3, we have,
The number of times of occurrence of 3 in prime factorisation of 6 = 1
The number of times of occurrence of 3 in prime factorisation of 8 = 0
The number of times of occurrence of 3 in prime factorisation of 9 = 2
So, the maximum number of times of occurrence of 3 = 2 …(ii)
Now using (i) and (ii), we can say that 2 must occur thrice, and 3 must occur twice. Thus,
$LCM=2\times 2\times 2\times 3\times 3$
$\Rightarrow LCM=72$

Hence, the LCM of 6, 8 and 9 is 72.

Note: Here, we can notice that the two numbers 8 and 9 do not have a common factor other than 1. So, the LCM of 8 and 9 will be $LCM=8\times 9=72$. We also see that 72 is divisible by 6. So, the LCM of 6, 8 and 9 will be 72.