To find the total number of comic books that Emma, Alex, and Michael have using the greatest common factor (GCF) and the distributive property, we can follow these steps:
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Find the GCF of the numbers of comic books:
- The numbers of comic books are:
- Emma: 63
- Alex: 49
- Michael: 77
- We find the prime factorization of each number:
- \(63 = 3^2 \times 7\)
- \(49 = 7^2\)
- \(77 = 7 \times 11\)
The GCF is the product of the lowest powers of the common prime factors. The only common factor here is \(7\).
- The numbers of comic books are:
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Use the GCF to express each number in terms of the GCF:
- Emma: \(63 = 7 \times 9\)
- Alex: \(49 = 7 \times 7\)
- Michael: \(77 = 7 \times 11\)
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Add these expressions using the distributive property:
- Total number of comic books = \(63 + 49 + 77\)
- This can be rewritten using the GCF: \[ 63 + 49 + 77 = 7 \times 9 + 7 \times 7 + 7 \times 11 \]
- Factor out the GCF (\(7\)): \[ = 7 \times (9 + 7 + 11) \]
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Calculate the sum inside the parentheses:
- \(9 + 7 + 11 = 27\)
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Multiply by the GCF: \[ = 7 \times 27 = 189 \]
Thus, the total number of comic books Emma, Alex, and Michael have is 189.