Question

The HCF of two numbers is $119$ and their LCM is $11781$. If one of the numbers is $1071$, find the other number?

Answer 1

Hint: In this problem we have given the values of HCF and LCM of the two numbers and one of the numbers also. We will assume the second number is $x$. Now we know that the product of the two numbers is equal to the product of the HCF and LCM of the two numbers.

Complete step-by-step solution
Given that, HCF is $119$ and LCM is $11781$.
We know that the product of HCF and LCM is given by
$\text{HCF}\times \text{LCM}=119\times 11781...\left( \text{i} \right)$
Given that one number is $1071$. Let the second number is $x$.
So, we will get the product of the two numbers is $1071\times x$.
We know that the product of HCF and LCM is equal to the product of the two numbers. Then
$\text{HCF}\times \text{LCM}=1071\times x$
Substituting the value of $\text{HCF}\times \text{LCM}$ from equation $\left( \text{i} \right)$, then we will get
$119\times 11781=1071\times x$
Dividing the above equation with $1071$ to get the value of $x$, then
$\begin{align}
  & \dfrac{119\times 11781}{1071}=\dfrac{1071\times x}{1071} \\
 & \Rightarrow x=1309 \\
\end{align}$
$\therefore $ The second number is $309$.

Note: We have used the principle that the product of HCF and LCM is equal to the product of the numbers. Here we will prove that by taking two numbers $18,45$. Factors of the two numbers are
$\begin{align}
  & 18=2\times 9 \\
 & \Rightarrow 18=2\times 3\times 3 \\
\end{align}$
$\begin{align}
  & 45=5\times 9 \\
 & \Rightarrow 45=5\times 3\times 3 \\
\end{align}$
LCM of the two numbers can be obtained by multiplying the factors of each number and if there are one or more factors repeated write them one time, then the LCM of $18,45$ is given by
$\begin{align}
  & \text{LCM}=2\times 3\times 5\times 3 \\
 & \Rightarrow \text{LCM}=90 \\
\end{align}$
HCF of the two numbers can be calculated by taking the common factors of two numbers. Then we will get
$\text{HCF}=3\times 3=9$
Now the product of LCM and HCF is
$\begin{align}
  & \text{HCF}\times \text{LCM}=9\times 90 \\
 & \Rightarrow \text{HCF}\times \text{LCM}=810 \\
\end{align}$
Now the product of two numbers is $18\times 45=810$
From this we can say that the product of HCF and LCM of the numbers is equal to the product of those numbers.