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LCM Formula

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  • Last Updated : 03 Nov, 2022
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The full form of LCM is Least Common Multiple. LCM of two numbers say a and b  is defined as the smallest positive integer which is divisible by both the numbers a and b. Hence, the LCM is a common multiple of two or more numbers. For example, LCM of 15 and 8 are 120. There are many methods of calculating the LCM of two or more numbers, which are explained below:

By Listing Multiples of Each Number

In this method, we need to list the multiples of each number until at least one of the multiples appears on all the lists. Then, the LCM is the smallest number that is on all of the lists. Example,

LCM of 6, 7, 21

Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60.

Multiples of 7 = 7, 14, 21, 28, 35, 42, 49, 56, 63, 70.

Multiples of 21 = 21, 42, 63, 84, 105, 126, 147, 168, 189, 210.

Now, the smallest number that is common in all the lists is 42.

Hence, 

LCM(6, 7, 21) = 42

To Find Out LCM using Prime Factorization Method

First, we need to write all numbers as a product of their prime factors. Then, LCM will be the product of the highest powers of all prime numbers. Example,

LCM of 15, 8

Prime factorization of 15 = 3 × 5

Prime factorization of 8 = 2 ×  2 × 2

Hence,

LCM = 23 × 3 × 5 = 120

To Find Out LCM using Division Method

First, we need to write all the numbers in a horizontal line separated by commas. Then, we divide all the given numbers by the smallest prime number. Then, we write the quotient and undivided numbers in a new line below the previous line. We repeat this procedure until we come to the stage where no prime factor is common. Then, we find the product of all divisors and the resultant number we get is the LCM. 

Example:

LCM of 6, 8, 5, 4, 3, 9 is 360

Findingoutlcmusingdivisionmethod

 

To Find LCM using GCD Formula

If a, and b are any numbers then, we know that,

LCM × HCF/GCD = a ×  b

LCM of a, b =  a × b / gcd(a,b)

Example: Find LCM of 4, 56 using GCD of 4, 56.

Solution:

Prime factors of 4 = 2 × 2

Prime factors of 56 = 2 × 2 × 2 × 7

Common factor = 2 × 2 = 4

Hence,

GCD of 4, 56 = 4

LCM of 4, 56 = (4 × 56)/ gcd of (4, 56)

= 224/ 4

= 56

LCM of Fractions

To find the LCM of two fractions we first compute the LCM of Numerators and GCD of the Denominators. Then, both these results will be expressed as a fraction 

LCM =  LCM of Numerators / GCD of Denominators    

Sample Problems on LCM Formula

Problem 1: Find out the LCM of 16 and 10.

Solution: 

we know that LCM(a, b) = a × b/ GCD(a, b)        

Here, a = 16 and b = 10

a × b = 16 × 10 = 160

GCD(a, b) = 2

Hence, LCM(16, 10) = 160 /2 = 80

Problem 2: Find the LCM of 6/7 and 5/4.

Solution: 

Numerators are 6, 5 and Denominators are 7, 4

Then, LCM(6, 5) = 30 

and GCD(7, 4) = 1

Hence, LCM of 6/7 and 5/4 = 30/1

Problem 3: Calculate the  LCM of 14, 12, 7, and 8.

Solution: 

Sampleproblemonlcmformula

 

LCM of 14, 12, 7, 8 = 2 × 2 × 2 × 3 × 7

                               = 168

Hence, LCM(14, 12, 7, 8) = 168

Problem 4: Find out the LCM for 8 and 24.

Solution: 

Prime Factorization of 8 = 2 × 2 × 2

Prime Factorization of 24 = 2 × 2 × 2 × 3

LCM = 2 × 2 × 2 = 8

Problem 5: Find out the LCM of 36, 4.

Solution:

Multiples of 36 = 36, 72, 108, 144, 180, 216, 252, 288, 324, 360 etc.

Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 etc.

Common multiple = 36

Hence, LCM of 36 and 4 = 36

Problem 6: Find the least number divided by 48 and 76, which leaves the remainder of 8 and 12 respectively.

Solution:

First we find the LCM of the two numbers we get,

Prime Factorisation of 48 = 2 × 2 × 2 × 2 × 3

Prime Factorization of 76 = 2 × 2 × 19

Therefore, LCM of the two numbers is 2 × 2 × 2 × 2 × 19 × 3 = 912.

The least number divided by 48 and 76 leaving remainder 8 and 12 is (912 – (8 + 12)) = 892.

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