LCM and HCF
L.C.M by Prime Factorisation Method:
To find the Least Common Multiple (LCM) of two or more numbers using the prime factorization method, you can follow these steps:
Let's find the LCM of 4 and 6 as an example
Step 1: Prime Factorization
Prime factorization of 4: 22 (because 2 x 2 = 4)
Prime factorization of 6: 21 x 31 (because 2 x 3 = 6)
Step 2: Identify the Prime Factors
List all the unique prime factors involved: 22 x 3
Step 3: Build the LCM
Multiply the highest power of each prime factor: 22 x 31 = 1
Step 4: Result
The LCM of 4 and 6 is 12
Let's take another example with three numbers: 4, 6, and 8
Step 1: Prime Factorization
Prime factorization of 4: 22 (because 2 x 2 = 4)
Prime factorization of 6: 21 x 31 (because 2 x 3 = 6)
Prime factorization of 8: 23 (because 2 x 2 x 2 = 8)
Step 2: Identify the Prime Factors
List all the unique prime factors involved: 23 x 3
Step 3: Build the LCM
Multiply the highest power of each prime factor: 23 x 31 = 24
Step 4: Result
The LCM of 4, 6, and 8 is 24
This method ensures that the LCM includes all the prime factors of each number with the highest power
Example I : Find the L.C.M of 15 and 21
Sol : \(15 = {3^1} \times {5^1},\) \(24 = 2 \times 2 \times 2 \times 3\) \(= {2^3} \times {3^1}\)
The product of numbers with highest powers
i.e 3 x 5 x 2 x 2 x 2 = 120
LCM = The product of numbers with highest powers
= 3 x 5 x 23 = 120
LCM and HCF
L.C.M by Prime Factorisation Method:
To find the Least Common Multiple (LCM) of two or more numbers using the prime factorization method, you can follow these steps:
Let's find the LCM of 4 and 6 as an example
Step 1: Prime Factorization
Prime factorization of 4: 22 (because 2 x 2 = 4)
Prime factorization of 6: 21 x 31 (because 2 x 3 = 6)
Step 2: Identify the Prime Factors
List all the unique prime factors involved: 22 x 3
Step 3: Build the LCM
Multiply the highest power of each prime factor: 22 x 31 = 1
Step 4: Result
The LCM of 4 and 6 is 12
Let's take another example with three numbers: 4, 6, and 8
Step 1: Prime Factorization
Prime factorization of 4: 22 (because 2 x 2 = 4)
Prime factorization of 6: 21 x 31 (because 2 x 3 = 6)
Prime factorization of 8: 23 (because 2 x 2 x 2 = 8)
Step 2: Identify the Prime Factors
List all the unique prime factors involved: 23 x 3
Step 3: Build the LCM
Multiply the highest power of each prime factor: 23 x 31 = 24
Step 4: Result
The LCM of 4, 6, and 8 is 24
This method ensures that the LCM includes all the prime factors of each number with the highest power
Example I : Find the L.C.M of 15 and 21
Sol : \(15 = {3^1} \times {5^1},\) \(24 = 2 \times 2 \times 2 \times 3\) \(= {2^3} \times {3^1}\)
The product of numbers with highest powers
i.e 3 x 5 x 2 x 2 x 2 = 120
LCM = The product of numbers with highest powers
= 3 x 5 x 23 = 120