Question

What is the LCM of 10, 15, 20 and 30?

Answer 1

Hint: This type of question depends on the concept of finding LCM that is least common multiple of the given numbers. When we have to find out the LCM, first write all the prime factors of each of the numbers. Then identify all factors with the highest number of occurrences and finally perform multiplication to obtain LCM.

Complete step by step answer:
Now, we have to find out the LCM of 10, 15, 20 and 30.
Initially, write the prime factorisation for each number
\[\begin{align}
  & \Rightarrow 10=2\times 5 \\
 & \Rightarrow 15=3\times 5 \\
 & \Rightarrow 20=2\times 2\times 5 \\
 & \Rightarrow 30=2\times 3\times 5 \\
\end{align}\]
Now we can observe that, the factor 2 occurs twice, the factor 3 occurs once and also the factor5 occurs only once. Hence, for the given numbers 10, 15, 20 and 30 LCM can be given by,
\[\Rightarrow LCM=2\times 2\times 3\times 5=60\]

Hence, LCM of 10, 15, 20 and 30 is 60.

Note: In this type of question one of the students may find the LCM by using common division method as follows:
First we write all the numbers in a horizontal line, separating them by using commas. Then, divide them by a suitable prime number which exactly divides at least two of the given numbers. We put the quotient directly under the numbers in the next row. If the number is not divided exactly, we bring it down in the next row. We continue the process until all one are left in the last row. Finally to obtain LCM we multiply all the prime numbers by which we divided.
\[\begin{align}
  & \Rightarrow 2\left| \!{\underline {\,
  10,15,20,30 \,}} \right. \\
 & \Rightarrow 5\left| \!{\underline {\,
  5,15,10,15 \,}} \right. \\
 & \Rightarrow 3\left| \!{\underline {\,
  1,3,2,3 \,}} \right. \\
 & \Rightarrow 2\left| \!{\underline {\,
  1,1,2,1 \,}} \right. \\
 & \Rightarrow 1\left| \!{\underline {\,
  1,1,1,1 \,}} \right. \\
\end{align}\]
\[\Rightarrow LCM=2\times 2\times 3\times 5=60\]