Question

The HCF of two numbers is $9$ and their LCM is $360$. If one of the numbers is $45$, find the other number?

Answer 1

Hint: When we are given with the HCF and LCM of the two numbers simply make an equation by using the known law i.e. the product of the HCF and LCM is equal to the product of the given numbers. So here we have given with the HCF and LCM of two number and also one of the number, so we will assume the second number as $x$ and put it in the equation that we have and calculates the value of $x$.

Complete step-by-step answer:
Given that, HCF is $9$ and LCM is $360$.
Now the product of the HCF and LCM is given by
$\text{HCF}\times \text{LCM}=9\times 360...\left( \text{i} \right)$
Given that one number is $45$. Let the second number is $x$.
Now the product of the two numbers is $45\times x$.
We know that the product of HCF and LCM is equal to the product of the two numbers. Then
$\text{HCF}\times \text{LCM}=45\times x$
Substituting the value of $\text{HCF}\times \text{LCM}$ from equation $\left( \text{i} \right)$, then we will get
$9\times 360=45\times x$
Dividing the above equation with $45$ to get the value of $x$, then
$\begin{align}
  & \frac{9\times 360}{45}=\frac{45\times x}{45} \\
 & \Rightarrow x=72 \\
\end{align}$
$\therefore $ The second number is $72$.

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 $24,36$. Factors of the two numbers are
$\begin{align}
  & 24=3\times 8 \\
 & \Rightarrow 24=3\times 2\times 4 \\
 & \Rightarrow 24=3\times 2\times 2\times 2 \\
\end{align}$ , $\begin{align}
  & 36=3\times 12 \\
 & \Rightarrow 36=3\times 3\times 4 \\
\end{align}$
LCM of the two numbers can be obtained by multiplying the least common factors of the both numbers i.e. multiply the factors which are smaller than other factors of that number, then the LCM of $24,36$ is given by
$\begin{align}
  & \text{LCM}=2\times 2\times 2\times 3\times 3 \\
 & \Rightarrow \text{LCM}=72 \\
\end{align}$
HCF of the two numbers can be calculated by multiplying the highest factors of the numbers then we will get
$\text{HCF}=3\times 4=12$
Now the product of LCM and HCF is
$\begin{align}
  & \text{HCF}\times \text{LCM}=12\times 72 \\
 & \Rightarrow \text{HCF}\times \text{LCM}=864 \\
\end{align}$
Now the product of two numbers is $24\times 36=864$
From this we can say that the product of HCF and LCM of the numbers is equal to the product of those numbers.