Question

Find the HCF and LCM of 36, 48 and 60 by Prime factorization method.

Answer 1

Hint: Find out prime factorization of 36, 48 and 60. Find the common factors in prime factorization of 36, 48 and 60.

Complete step-by-step answer:
The product of all of their prime factors will be HCF, To find the LCM, find the product of HCF and all the remaining prime factors of 36, 48 and 60.

We have to find HCF and LCM of 36, 48 and 60 by prime factorization method.
Let us first find out prime factorization of 36, 48 and 60. One by one.
Prime factorization of 36,
\[\begin{align}
  & 2\left| \!{\underline {\,
  36 \,}} \right. \\
 & 2\left| \!{\underline {\,
  18 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ }1 \\
\end{align}\]
$\Rightarrow 36=2\times 2\times 3\times 3$
Prime factorization of 48,
$\begin{align}
  & 2\left| \!{\underline {\,
  48 \,}} \right. \\
 & 2\left| \!{\underline {\,
  24 \,}} \right. \\
 & 2\left| \!{\underline {\,
  12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \text{ 1} \\
 & \Rightarrow \text{48=2}\times \text{2}\times \text{2}\times \text{2}\times \text{3} \\
\end{align}$
Prime factorization of 60.
$\begin{align}
  & 2\left| \!{\underline {\,
  60 \,}} \right. \\
 & 2\left| \!{\underline {\,
  30 \,}} \right. \\
 & 3\left| \!{\underline {\,
  15 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & \text{ 1} \\
 & \Rightarrow \text{60=2}\times \text{2}\times \text{3}\times \text{5} \\
\end{align}$
We have got,
$\begin{align}
  & 36=2\times 2\times 3\times 3 \\
 & 48=2\times 2\times 2\times 2\times 3 \\
 & 60=2\times 2\times 3\times 5 \\
\end{align}$
Let us first find out the HCF of 36, 48 and 60.
To find the HCF of three numbers, we have to find the factors which one common in all three and HCF will be the product of all common factors of these three numbers
$\begin{align}
  & 36=2\times 2\times 3\times 3 \\
 & 48=2\times 2\times 2\times 2\times 3 \\
 & 60=2\times 2\times 3\times 5 \\
\end{align}$
HCF$=2\times 2\times 3$ . Because 2,2 and 3 are the common factors in all three numbers 36, 48 and 60
$\Rightarrow HCF=12$
So, the HCF of 36, 48 and 60 is 12.
Now, let us find out LCM of 36, 48 and 60. To find the LCM, we multiply HCF with the remaining factors which are not common. We can see above that 3,2,2 and 5 are remaining. So, $LCM=HCF\times \left( 3\times 2\times 2\times 5 \right)$
$\begin{align}
  & \Rightarrow LCM=12\times 3\times 2\times 2\times 5 \\
 & \Rightarrow LCM=720 \\
\end{align}$
Hence LCM and HCF of 36, 48 and 60 are 720 and 12 respectively.

Note: Another method to find LCM of 36, 48 and 60,
$\begin{align}
  & 2\left| \!{\underline {\,
  36,48,60 \,}} \right. \\
 & 2\left| \!{\underline {\,
  18,24,30 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9,12,15 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  3,4,5 \,}} \right. \\
 & LCM=2\times 2\times 3\times 3\times 4\times 5 \\
 & =720 \\
\end{align}$