Question

Find the HCF and LCM of $306$ and $657$ by prime factorization method.

Answer 1

Hint: In this method we need to find the HCF and LCM of the given two numbers. The terms HCF and LCM stand for Highest Common Factor and Lowest Common Multiple. In order to find HCF or LCM we need to have the given numbers in the prime factored form. From the prime factored form, we will simplify the possible factors by converting them into exponential form. Now the HCF of the two numbers is obtained by multiplying the common factors of the two numbers and the LCM of the two numbers is obtained by multiplying the remaining factors and common factors one time of the two numbers.

Complete step by step answer:
Given numbers are $306$ and $657$.
Considering the number $306$. The prime factored form of the number $306$ is given by
$\begin{align}
  & 2\left| \!{\underline {\,
  306 \,}} \right. \\
 & 3\left| \!{\underline {\,
  153 \,}} \right. \\
 & 3\left| \!{\underline {\,
  51 \,}} \right. \\
 & 17\left| \!{\underline {\,
  17 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
From this we can write the number $306$ as
$306=2\times 3\times 3\times 17$
Using the formula $a\times a\times a\times a....\text{ n times}={{a}^{n}}$ to write the above value in exponential form, then we will get
$306=2\times {{3}^{2}}\times 17$
Considering the number $657$. The prime factored form of the number $657$ is given by
$\begin{align}
  & 3\left| \!{\underline {\,
  657 \,}} \right. \\
 & 3\left| \!{\underline {\,
  219 \,}} \right. \\
 & \left| \!{\underline {\,
  73 \,}} \right. \\
\end{align}$
From this we can write the number $657$ as
$\begin{align}
  & 657=3\times 219 \\
 & \Rightarrow 657=3\times 3\times 73 \\
\end{align}$
Using the formula $a\times a\times a\times a....\text{ n times}={{a}^{n}}$ to write the above value in exponential form, then we will get
$657={{3}^{2}}\times 73$
We can observe that the common factors in both the given numbers is ${{3}^{2}}$.
So, the HCF of the given two numbers is ${{3}^{2}}=9$.
Now the LCM of the two numbers is obtained by multiplying the remaining factors and the common factors at one time of the both the numbers, then we will get
$\begin{align}
  & \text{LCM}=2\times {{3}^{2}}\times 17\times 73 \\
 & \Rightarrow \text{LCM}=22338 \\
\end{align}$

Hence the HCF and LCM of the numbers $306$ and $657$ are $9$ and $22338$ respectively.

Note: LCM of the numbers is also calculated by using the relation between the LCM, HCF and the given number. The product of the LCM and HCF is equal to the Product of the given numbers. We can use the above rule after finding the value of HCF to find the value of LCM.