find the hcf of 18,27 and 54 by prime factorisation method
The Correct Answer and Explanation is:
Correct Answer:
The HCF (Highest Common Factor) of 18, 27, and 54 using the prime factorization method is 9.
Step-by-Step Explanation:
1. Prime Factorization of Each Number:
- 18 = 2 × 3 × 3 = 2 × 3²
- 27 = 3 × 3 × 3 = 3³
- 54 = 2 × 3 × 3 × 3 = 2 × 3³
2. Identify Common Prime Factors:
All three numbers have at least one common factor: 3.
The lowest power of 3 in the prime factorization of all three numbers is 3² (which is 9).
Even though 2 appears in 18 and 54, it does not appear in 27. So, it is not included in the HCF.
3. Multiply the Common Prime Factors:
- Only 3² is common in all, and no other primes are common.
- Therefore, HCF = 3² = 9
Detailed Explanation (300 words):
The Highest Common Factor (HCF), also known as the greatest common divisor, is the largest positive integer that divides each of the given numbers without leaving a remainder. To determine the HCF of 18, 27, and 54 using the prime factorization method, each number must be broken down into its prime components.
Start with 18. Divide by the smallest prime number, 2. Since 18 ÷ 2 = 9, continue with 9. The number 9 is divisible by 3, giving 3. Dividing again by 3 gives 1. So, 18 = 2 × 3².
Now factor 27. It is not divisible by 2 but divisible by 3. Repeated division by 3 yields 27 → 9 → 3 → 1. So, 27 = 3³.
Next, factor 54. Divide by 2 first: 54 ÷ 2 = 27. Then proceed with 3 repeatedly until reaching 1. The complete factorization is 2 × 3³.
After listing the prime factors, identify those common to all three numbers. The only prime common to all is 3. The smallest exponent of 3 among the factorizations is 2 (from 18). Therefore, the HCF is obtained by multiplying 3 raised to the smallest power: 3² = 9.
This method ensures that the largest shared factor is found by considering only the lowest powers of the common prime numbers, resulting in the correct HCF.
