To find the greatest common factor (GCF) of 126 and 98, we can start by determining the prime factorizations of both numbers.
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Finding the prime factorization of 126:
- 126 is even, so divide by 2: \( 126 \div 2 = 63 \)
- Next, factor 63: \( 63 = 3 \times 21 \)
- Factor 21: \( 21 = 3 \times 7 \)
- Thus, the prime factorization of 126 is: \[ 126 = 2^1 \times 3^2 \times 7^1 \]
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Finding the prime factorization of 98:
- 98 is even, so divide by 2: \( 98 \div 2 = 49 \)
- Factor 49: \( 49 = 7 \times 7 = 7^2 \)
- Thus, the prime factorization of 98 is: \[ 98 = 2^1 \times 7^2 \]
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Identifying the common factors:
- From the prime factorizations:
- 126: \( 2^1 \times 3^2 \times 7^1 \)
- 98: \( 2^1 \times 7^2 \)
The common prime factors are \( 2^1 \) and \( 7^1 \).
- From the prime factorizations:
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Calculating the GCF:
- Multiply the lowest powers of the common prime factors: \[ \text{GCF} = 2^1 \times 7^1 = 2 \times 7 = 14 \]
Therefore, the greatest common factor of 126 and 98 is \( \boxed{14} \).