LCM Important Formulas, Definitions, & Examples:


Overview:


Topic Included:Formulas, Definitions & Exmaples.
Main Topic:Quantitative Aptitude.
Quantitative Aptitude Sub-topic:Number System Aptitude Notes & Questions.
Questions for practice:10 Questions & Answers with Solutions.

What is LCM?


The LCM stands for Least Common Multiple, it is also referred to as Lowest Common Multiple or Smallest Common Multiple. The LCM of two numbers a and b is the smallest common multiple of two integers which is exactly divisible by each of them a and b. LCM is not defined for negative numbers and zero.


LCM Examples:


What is the LCM of 6 and 8?


Answer: The multiples of 6 are \(= 6 \times 12 \times 18 \times 24....\) and multiples of 8 are \(= 8 \times 16 \times 24....\). Hence the LCM of 6 and 8 is 24. Here 24 is the smallest common multiple which is exactly divisible by 6 and 8.


What is the LCM of 5 and 7?


Answer: The multiples of 5 are \(= 5 \times 10 \times 15 \times 20 \times 25 \times 30 \times 35....\) and multiples of 7 are \(= 7 \times 14 \times 21 \times 28 \times 35....\). Hence the LCM of 5 and 7 is 35. Here 35 is the smallest common multiple which is exactly divisible by 5 and 7.


What is the LCM of 4 and 6?


Answer: The multiples of 4 are \(= 4 \times 8 \times 12....\) and multiples of 6 are \(= 6 \times 12....\). Hence the LCM of 4 and 6 is 12. Here 12 is the smallest common multiple which is exactly divisible by 4 and 6.


What is the LCM of 6 and 9?


Answer: The multiples of 6 are \(= 6 \times 12 \times 18....\) and multiples of 9 are \(= 9 \times 18....\). Hence the LCM of 6 and 9 is 18. Here 18 is the smallest common multiple which is exactly divisible by 6 and 9.


How to find LCM of three Numbers?


The LCM of smaller numbers can be found easily but, what if we need to find the LCM of bigger numbers. We can follow the process given below to find the LCM of bigger numbers and the LCM of three numbers.


Step (1): First, Factorize the numbers into their prime factors.

Step (2): Collect all the distinct factors with their maximum available power.

Step (3): Multiply the collected factors to get the LCM.


Example to find LCM of three Numbers:


What is the LCM of 12, 15, and 18?


Solution: Step(1): Factorize the numbers into their prime factors.$$ 12 = 2 \times 2 \times 3 = 2^{2} \times 3^{1} $$ $$ 15 = 3 \times 5 = 3^{1} \times 5^{1} $$ $$ 18 = 2 \times 3 \times 3 = 2^{1} \times 3^{2} $$ Step(2): Collect all the distinct factors with their maximum available power.$$ = 2^{2}, \ 3^{2}, \ 5^{1} $$ Step(3): Multiply the collected factors.$$ = 2^{2} \times 3^{2} \times 5^{1} $$ $$ = 4 \times 9 \times 5 = 180 $$ Here, 180 is the smallest positive number which is exactly divisible by 12, 15, and 18.


How to find LCM of fractions?


We know that fractions always have two parts, Numerator and Denominator, and written in the form of \(\frac{Numerator}{Denominator}\). Where the denominator should not be zero. The LCM of fractions can be found by using the below formula.


Formula to find LCM of a Fraction:


$$ LCM \ of \ fractions = \frac{LCM \ of \ Numerators}{HCF \ of \ Denominators} $$

\(LCM \ of \ fractions = \frac{LCM \ of \ Numerators}{HCF \ of \ Denominators}\)


LCM of Fractions Examples:


What is the LCM of \(\frac{4}{5}\) and \(\frac{5}{6}\)?


Solution:

$$ LCM \ of \ fractions = \frac{LCM \ of \ Numerators}{HCF \ of \ Denominators} $$

\(LCM \ of \ fractions = \frac{LCM \ of \ Numerators}{HCF \ of \ Denominators}\)

$$ LCM = \frac{LCM \ of \ (4,5)}{HCF \ of \ (5,6)} $$ $$ = \frac{20}{1} = 20 $$


What is the LCM of \(\frac{2}{3}\) and \(\frac{3}{4}\)?


Solution:

$$ LCM \ of \ fractions = \frac{LCM \ of \ Numerators}{HCF \ of \ Denominators} $$

\(LCM \ of \ fractions = \frac{LCM \ of \ Numerators}{HCF \ of \ Denominators}\)

$$ LCM = \frac{LCM \ of \ (2,3)}{HCF \ of \ (3,4)} $$ $$ = \frac{6}{1} = 6 $$


More LCM Examples:


What is the LCM of 9 and 12?


Answer: The multiples of 9 are \(= 9 \times 18 \times 27 \times 36....\) and multiples of 12 are \(= 12 \times 24 \times 36....\). Hence the LCM of 9 and 12 is 36. Here 36 is the smallest common multiple which is exactly divisible by both 9 and 12.


What is the LCM of 8 and 12?


Answer: The multiples of 8 are \(= 8 \times 16 \times 24....\) and multiples of 12 are \(= 12 \times 24....\). Hence the LCM of 8 and 12 is 24. Here 24 is the smallest common multiple which is exactly divisible by both 8 and 12.


What is the LCM of 8 and 10?


Answer: The multiples of 8 are \(= 8 \times 16 \times 24 \times 32 \times 40....\) and multiples of 10 are \(= 10 \times 20 \times 30 \times 40....\). Hence the LCM of 8 and 10 is 40. Here 40 is the smallest common multiple which is exactly divisible by both 8 and 10.


Practice Unsolved Questions


What is the LCM of 3 and 6?

What is the LCM of 4 and 10?

What is the LCM of 4 and 9?

What is the LCM of 7 and 11?

What is the LCM of 10 and 11?

What is the LCM of 10 and 14?

What is the LCM of 10 and 15?

What is the LCM of 10 and 16?

What is the LCM of 16 and 20?

What is the LCM of 12 and 11?

What is the LCM of 4 and 12?

What is the LCM of 3, 4 and 5?

What is the LCM of 5, 6 and 7?

What is the LCM of 10, 11 and 12?

What is the LCM of 11 and 44?

What is the LCM of 11 and 9?

What is the LCM of 5 and 9?

What is the LCM of 33 and 44?