Find the H.C.F of the following numbers using prime factorization method 150, 140, 210.

To find the Highest Common Factor (HCF) of 150, 140, and 210 using the prime factorization method, we follow these steps:

Step 1: Prime Factorization of 150
Start by dividing 150 by the smallest prime number (2, 3, 5, etc.):

150 ÷ 2 = 75
75 ÷ 3 = 25
25 ÷ 5 = 5
5 ÷ 5 = 1
So, the prime factorization of 150 is: 150 = 2 × 3 × 5²

Step 2: Prime Factorization of 140
Now, perform the prime factorization of 140:

140 ÷ 2 = 70
70 ÷ 2 = 35
35 ÷ 5 = 7
7 ÷ 7 = 1
So, the prime factorization of 140 is: 140 = 2² × 5 × 7

Step 3: Prime Factorization of 210
Next, perform the prime factorization of 210:

210 ÷ 2 = 105
105 ÷ 3 = 35
35 ÷ 5 = 7
7 ÷ 7 = 1
So, the prime factorization of 210 is: 210 = 2 × 3 × 5 × 7

Step 4: Common Prime Factors
The common prime factors among 150, 140, and 210 are 2 and 5.

Step 5: Find the Lowest Powers of Common Prime Factors
The lowest power of 2 is 2¹ (since 150 has 2¹, 140 has 2², and 210 has 2¹).
The lowest power of 5 is 5¹ (since all three numbers have 5¹).
Step 6: Calculate the HCF
Now, multiply the lowest powers of the common prime factors: HCF = 2¹ × 5¹ = 2 × 5 = 10

Final Answer:
The HCF of 150, 140, and 210 is 10.