Find the HCF of the following numbers, using the prime factorization method: 272, 425
The correct answer and explanation is:
To find the Highest Common Factor (HCF) of 272 and 425 using the prime factorization method, follow these steps:
Step 1: Prime Factorization of 272
Start by dividing 272 by the smallest prime number, which is 2.
272 ÷ 2 = 136
136 ÷ 2 = 68
68 ÷ 2 = 34
34 ÷ 2 = 17
17 is a prime number and cannot be divided further.
So, the prime factorization of 272 is:
272 = 2⁴ × 17
Step 2: Prime Factorization of 425
Next, divide 425 by the smallest prime number, which is 5.
425 ÷ 5 = 85
85 ÷ 5 = 17
17 is a prime number and cannot be divided further.
So, the prime factorization of 425 is:
425 = 5² × 17
Step 3: Identify Common Factors
The prime factors of 272 are 2⁴ × 17, and the prime factors of 425 are 5² × 17. The only common prime factor between 272 and 425 is 17.
Step 4: Determine the HCF
The HCF is the product of the lowest powers of all common prime factors. In this case, the only common prime factor is 17, and its lowest power is 17ⁱ where i = 1.
Therefore, the HCF of 272 and 425 is 17.
Conclusion
The Highest Common Factor of 272 and 425 is 17. This method involves breaking down each number into its prime factors, identifying the common factors, and then multiplying the lowest powers of those common factors to find the HCF.