Question

Find the LCM of 28, 36 and 60.

Answer 1

Hint: To find the LCM of given three numbers, first we will write the given numbers in terms of multiplication of prime numbers and then we will group those numbers which are common. Then we will take one number from the group and we will multiply to the rest of the ungrouped numbers.

Complete step-by-step answer:
Before starting to solve the question, we must first know what LCM is. LCM is the short form of lowest common multiple or least common multiple and it is defined as the smallest positive integer that is divisible by all the numbers of which we have to find the LCM. The LCM of three numbers is calculated as shown in the following steps.
Step I: we will first resolve the given numbers in terms of multiplication of prime factors i.e. we have to find prime factors of the given numbers. The prime factors of 28 are:
$\begin{align}
  & 2\left| \!{\underline {\,
  28 \,}} \right. \\
 & 2\left| \!{\underline {\,
  14 \,}} \right. \\
 & 7\left| \!{\underline {\,
  7 \,}} \right. \\
 & 1 \\
\end{align}$
The prime factors of 36 are:
$\begin{align}
  & 2\left| \!{\underline {\,
  36 \,}} \right. \\
 & 2\left| \!{\underline {\,
  18 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & 1 \\
\end{align}$
The prime factors of 60 are:
$\begin{align}
  & 2\left| \!{\underline {\,
  60 \,}} \right. \\
 & 2\left| \!{\underline {\,
  30 \,}} \right. \\
 & 3\left| \!{\underline {\,
  15 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & 1 \\
\end{align}$
Therefore, we can say that:
$28=2\times 2\times 7$ .
$36=2\times 2\times 3\times 3$ .
$60=2\times 2\times 3\times 5$ .
Step II: we will find those numbers which are common all the three numbers or which are common to any two numbers. In our case we can see that numbers which are common to all the three numbers are 2 and 2. Also, 3 is common to both the 36 and 60. So the groups we get are = $\left( 2\times 2\times 2 \right),\left( 2\times 2\times 2 \right)$ and $\left( 3\times 3 \right)$ .
Step III: we will now take one factor from each group and then we will multiply them with ungrouped numbers. The LCM of the number will be the product obtained. The LCM $\left( 28,36,60 \right)=2\times 2\times 3\times 7\times 3\times 5$
$=1260.$

Note: The LCM of the three numbers can also be calculated as shown below
$\begin{align}
  & 2\left| \!{\underline {\,
  28,36,60 \,}} \right. \\
 & 2\left| \!{\underline {\,
  14,18,30 \,}} \right. \\
 & 3\left| \!{\underline {\,
  7,9,15 \,}} \right. \\
 & 3\left| \!{\underline {\,
  7,3,5 \,}} \right. \\
 & 5\left| \!{\underline {\,
  7,1,5 \,}} \right. \\
 & 7\left| \!{\underline {\,
  7,1,1 \,}} \right. \\
 & 1,1,1 \\
\end{align}$
LCM $=2\times 2\times 3\times 3\times 5\times 7$
$=1260.$