Question

What is the least common multiple of $8$ and $12$?

Answer 1

Hint: For finding out the LCM of the given two numbers, which are $8$ and $12$, we need to consider the prime factorization of these two numbers. For the prime factorization, we have to use the long division method. After the prime factorization, we will write both the numbers as the multiplication of their respective prime factors. Then finally, to get the LCM, we have to multiply the common factors of the two numbers with the uncommon factors.

Complete step-by-step solution:
The two numbers given in the above question are $8$ and $12$. According to the question, we have to find the least common multiple of these two numbers. For this, we have to write these two numbers as the multiplication of their prime factors. Therefore, we consider the prime factorization of these two numbers which can be done by using the long division method. The prime factorization of the number $8$ is shown below.
\[\begin{align}
  & 2\left| \!{\underline {\,
  8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
Similarly, we do the prime factorization of the number $12$ as shown below.
\[\begin{align}
  & 2\left| \!{\underline {\,
  12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
Therefore, we can write the given two numbers as
$\begin{align}
  & \Rightarrow 8=2\times 2\times 2 \\
 & \Rightarrow 12=2\times 2\times 3 \\
\end{align}$
We see in the above expressions that the factor $2\times 2$ is common to both the numbers. And the remaining factors which are not common to the two numbers are $2$ and $3$. For obtaining the LCM of the given numbers, we have to multiply the common factors with the uncommon factors. Therefore, the LCM of the numbers $8$ and $12$ is given by
\[\begin{align}
  & \Rightarrow LCM\left( 8,12 \right)=2\times 2\times 2\times 3 \\
 & \Rightarrow LCM\left( 8,12 \right)=24 \\
\end{align}\]
Hence, the LCM of the given numbers $8$ and $12$ is equal to \[24\].

Note: Do not consider the LCM as simply the multiplication of the factors common to the numbers. The product of the common factors is equal to the highest common factor, or the HCF. Also, do not understand the LCM as the multiplication of the numbers. The multiplication of two numbers is the common multiple to the two numbers, but it is not necessarily the least common multiple.