Question

What is the LCM of 9 and 12?

Answer 1

Hint: LCM or Least Common Multiple of a and b can be defined as the smallest number that is divisible by both a and b. To calculate the LCM of two numbers, we should first find the prime factorisation of both numbers. 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 solution:
Least Common Multiple (LCM) of a and b is defined as the smallest number possible that is perfectly divisible by both a and b.
The most common method to find LCM is the Prime Factorisation Method.
In this method, we should first calculate the prime factorisation, and then list the prime factors as many times as the maximum number of times of their occurrence.
Here, we have to calculate the LCM of 9 and 12.
Prime factorisation of 9:
\[\begin{align}
  & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
$\therefore 9=3\times 3$
Prime factorisation of 12:
$\begin{align}
  & 2\left| \!{\underline {\,
  12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
$\therefore 12=2\times 2\times 3$
For the prime factor 2, we have,
Number of times of occurrence of 2 in prime factorisation of 9 = 0
Number of times of occurrence of 2 in prime factorisation of 12 = 2
Thus, we can clearly see that,
Maximum number of times of occurrence of 2 = 2 …(i)
Similarly, for prime factor 3, we have,
Number of times of occurrence of 3 in prime factorisation of 9 = 2
Number of times of occurrence of 3 in prime factorisation of 12 = 1
Thus, we can clearly see that,
Maximum number of times of occurrence of 3 = 2 …(ii)
Now using (i) and (ii), we can say that 2 must occur twice, and 3 must also occur twice. Thus,
$LCM=2\times 2\times 3\times 3$
$\Rightarrow LCM=36$
Hence, the LCM of 9 and 12 is 36.

Note: We can also use the repeated division method to find the LCM, which we have shown below:
$\begin{align}
  & 2\left| \!{\underline {\,
  9,12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  9,6\text{ } \,}} \right. \\
 & 3\left| \!{\underline {\,
  9,3\text{ } \,}} \right. \\
 & 3\left| \!{\underline {\,
  3,1\text{ } \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1,1\text{ } \,}} \right. \\
\end{align}$
$\therefore LCM=2\times 2\times 3\times 3=36$