LCM Formula

Let is discuss different ways to compute LCM using direct formula.

LCM using HCF Formula

LCM can also be derived using GCD or HCF. If a, and b are any numbers then, we have,

LCM(a,b) × HCF(a,b) = a ×  b

or

LCM(a,b) =  (a × b)/ GCD(a,b)

Note: This formula is only applicable while finding the LCM of two numbers only.

Example: Find LCM of 4and 56 using their GCD

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 and 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. The formula to calculate LCM of two fractions is given below:

LCM =  LCM of Numerators / GCD of Denominators    

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

Solution: 

Numerators are 6, 5 and Denominators are 7, 4

We know that LCM of Fractions = LCM of Numerators / GCD of Denominators  

Then, LCM(6, 5) = 30 

and GCD(7, 4) = 1

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

LCM Formula Examples

Example 1: Find out the LCM of 4 and 10.

Solution: 

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

Here, a = 4 and b = 10
a × b = 4 × 10 = 40
GCD(a, b) = 2

Hence, LCM(16, 10) = 40 /2 = 20

Example 2: Calculate the  LCM of 14, 12, 7, and 8.

Solution: 

LCM of 14, 12, 7, 8 = 2 × 2 × 2 × 3 × 7 = 168

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

Example 3: 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 × 3= 24

Example 4: Find out the LCM of 36 and 24.

Solution:

Multiples of 36 = 36, 72, 108, 144, 180, 216, 252, 288, 324, 360 etc.
Multiples of 24 = 24, 48, 72, 96, 120, etc.

Common multiple = 72

Hence, LCM of 36 and 24 = 72

Example 5: Find the least number divided by 48 and 72, which leaves the remainder 9 in each.

Solution:

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

Prime Factorisation of 48 = 2 × 2 × 2 × 2 × 3
Prime Factorization of 72 = 2 × 2 × 2 × 3 × 3

Therefore, LCM of the two numbers is 2 × 2 × 2 × 2 × 3 × 3 = 144.

The least number divided by 48 and 76 leaving remainder 9 is (144 + 9) = 153.

Practice Questions on LCM Formula

Question 1: Find the LCM of 22, 25, 11 and 8 by division method.

Question 2: If HCF(12, 24) = 12. Find LCM(12, 24).

Question 3: Find the LCM of 2/3 and 3/4.

Question 4: Prove that HCF(24, 48) × LCM(24, 48) = 24 × 48.

Question 5: Find the least number divided by 36 and 24, that leaves the remainder 3 in each case.

Read More: Practice Questions on LCM