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Answer :
To find the Greatest Common Factor (GCF) of 56 and 84, we can start by determining the prime factorization of each number.
1. Prime Factorization of 56:
- 56 is even, so it's divisible by 2: [tex]\(56 \div 2 = 28\)[/tex].
- 28 is even, so it's divisible by 2: [tex]\(28 \div 2 = 14\)[/tex].
- 14 is even, so it's divisible by 2: [tex]\(14 \div 2 = 7\)[/tex].
- 7 is a prime number.
Therefore, the prime factorization of 56 is [tex]\(2^3 \times 7\)[/tex].
2. Prime Factorization of 84 (using the factor tree from the question):
- Start with the given tree, where 84 breaks into 2 and another factor.
- 84 is divisible by 2: [tex]\(84 \div 2 = 42\)[/tex].
- 42 is divisible by 2: [tex]\(42 \div 2 = 21\)[/tex].
- 21 is divisible by 3: [tex]\(21 \div 3 = 7\)[/tex].
- 7 is a prime number.
Therefore, the prime factorization of 84 is [tex]\(2^2 \times 3 \times 7\)[/tex].
3. Finding the GCF:
- Compare the prime factorizations:
- For 2, the smallest power of 2 common to both numbers is [tex]\(2^2\)[/tex].
- For 7, both numbers have a factor of 7.
Multiply the smallest powers of all common prime factors: [tex]\(2^2 \times 7 = 4 \times 7 = 28\)[/tex].
Thus, the GCF of 56 and 84 is 28.
1. Prime Factorization of 56:
- 56 is even, so it's divisible by 2: [tex]\(56 \div 2 = 28\)[/tex].
- 28 is even, so it's divisible by 2: [tex]\(28 \div 2 = 14\)[/tex].
- 14 is even, so it's divisible by 2: [tex]\(14 \div 2 = 7\)[/tex].
- 7 is a prime number.
Therefore, the prime factorization of 56 is [tex]\(2^3 \times 7\)[/tex].
2. Prime Factorization of 84 (using the factor tree from the question):
- Start with the given tree, where 84 breaks into 2 and another factor.
- 84 is divisible by 2: [tex]\(84 \div 2 = 42\)[/tex].
- 42 is divisible by 2: [tex]\(42 \div 2 = 21\)[/tex].
- 21 is divisible by 3: [tex]\(21 \div 3 = 7\)[/tex].
- 7 is a prime number.
Therefore, the prime factorization of 84 is [tex]\(2^2 \times 3 \times 7\)[/tex].
3. Finding the GCF:
- Compare the prime factorizations:
- For 2, the smallest power of 2 common to both numbers is [tex]\(2^2\)[/tex].
- For 7, both numbers have a factor of 7.
Multiply the smallest powers of all common prime factors: [tex]\(2^2 \times 7 = 4 \times 7 = 28\)[/tex].
Thus, the GCF of 56 and 84 is 28.
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