To find the Highest Common Factor (H.C.F.), also known as the Greatest Common Divisor (GCD), of given pairs of numbers, please follow the step-by-step process below:
1. H.C.F. of 74 and 60:
- Prime Factorization of 74:
- 74 is divisible by 2 (the smallest prime number), so: 74 ÷ 2 = 37
- 37 is a prime number.
- Therefore, the prime factorization of 74 is: [tex]\(74 = 2 \times 37\)[/tex]
- Prime Factorization of 60:
- 60 is divisible by 2: 60 ÷ 2 = 30
- 30 is also divisible by 2: 30 ÷ 2 = 15
- 15 is divisible by 3: 15 ÷ 3 = 5
- 5 is a prime number.
- Therefore, the prime factorization of 60 is: [tex]\(60 = 2^2 \times 3 \times 5\)[/tex]
- Common Factors of 74 and 60:
- The only common prime factor between 74 and 60 is 2.
- H.C.F. of 74 and 60:
- The highest common factor is 2.
2. H.C.F. of 30 and 43:
- Prime Factorization of 30:
- 30 is divisible by 2: 30 ÷ 2 = 15
- 15 is divisible by 3: 15 ÷ 3 = 5
- 5 is a prime number.
- Therefore, the prime factorization of 30 is: [tex]\(30 = 2 \times 3 \times 5\)[/tex]
- Prime Factorization of 43:
- 43 is a prime number since it is not divisible by any number other than itself and 1.
- Therefore, the prime factorization of 43 is: [tex]\(43 = 43\)[/tex]
- Common Factors of 30 and 43:
- There are no common prime factors between 30 and 43.
- H.C.F. of 30 and 43:
- The highest common factor is 1 (since no common prime factor exists).
Therefore, the H.C.F. of the pairs 74 and 60, and 30 and 43 are:
- H.C.F. of 74 and 60 is: 2
- H.C.F. of 30 and 43 is: 1
