Find the hcf of 210 140 150 using prime factorization method
The Correct Answer and Explanation is:
To find the HCF (Highest Common Factor) of 210, 140, and 150 using the prime factorization method, follow these steps:
Step 1: Prime Factorization
210
210 ÷ 2 = 105
105 ÷ 3 = 35
35 ÷ 5 = 7
7 ÷ 7 = 1
→ Prime factorization of 210 = 2 × 3 × 5 × 7
140
140 ÷ 2 = 70
70 ÷ 2 = 35
35 ÷ 5 = 7
7 ÷ 7 = 1
→ Prime factorization of 140 = 2² × 5 × 7
150
150 ÷ 2 = 75
75 ÷ 3 = 25
25 ÷ 5 = 5
5 ÷ 5 = 1
→ Prime factorization of 150 = 2 × 3 × 5²
Step 2: Identify Common Prime Factors
Now list the common prime factors from all three factorizations:
- 2 is present in all three numbers
- 5 is present in all three numbers
- 3 is not in 140
- 7 is not in 150
Thus, the common prime factors = 2 and 5
Now consider the lowest powers of these common primes:
- 2 appears as 2¹ in 210 and 150, and 2² in 140 → take 2¹
- 5 appears as 5¹ in 210 and 140, and 5² in 150 → take 5¹
Step 3: Multiply the Common Prime Factors
HCF = 2¹ × 5¹ = 2 × 5 = 10
Final Answer: 10
Explanation
The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), represents the largest positive number that divides all the given numbers without leaving a remainder. The prime factorization method is a systematic approach to identify this number.
To apply this method, each number must first be broken down into its prime components. Prime numbers are numbers greater than 1 that are divisible only by 1 and themselves. By repeatedly dividing the number by the smallest possible prime numbers, a product of primes is obtained. This process continues until the result is 1.
Once all numbers are factorized, the next step is to examine the prime factors of each and identify the common ones. It is important to only consider the prime numbers that appear in all the factorizations. The HCF is then determined by multiplying these common primes using their smallest powers found in the individual factorizations.
For the numbers 210, 140, and 150, the prime factorizations are 2 × 3 × 5 × 7, 2² × 5 × 7, and 2 × 3 × 5² respectively. The only primes that appear in all three are 2 and 5. Among these, the lowest power of 2 is 2¹ and the lowest power of 5 is 5¹. Multiplying them results in 2 × 5 = 10, which is the largest number that divides all three numbers exactly.
This method is especially useful for large numbers and ensures that no common factor is missed. It also helps reinforce an understanding of prime numbers and their role in number theory.
