To find the greatest common factor (GCF) of the numbers 75, 66, 63, and 48, we first need to determine the prime factorization of each number:
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Prime factorization of 75: \[ 75 = 3 \times 5^2 \]
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Prime factorization of 66: \[ 66 = 2 \times 3 \times 11 \]
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Prime factorization of 63: \[ 63 = 3^2 \times 7 \]
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Prime factorization of 48: \[ 48 = 2^4 \times 3 \]
Now, we will find the common prime factors among these numbers. The prime factors we identified are:
- 75: \(3, 5\)
- 66: \(2, 3, 11\)
- 63: \(3, 7\)
- 48: \(2, 3\)
Now, the only prime factor that appears in all four factorizations is \(3\).
Next, we will determine the minimum exponent of this common prime factor (which is \(3\)) across all the factorizations. The minimum exponent for the prime \(3\) is \(1\) since:
- In 75, the exponent of \(3\) is \(1\) (i.e., \(3^1\)).
- In 66, the exponent of \(3\) is \(1\) (i.e., \(3^1\)).
- In 63, the exponent of \(3\) is \(2\) (i.e., \(3^2\)).
- In 48, the exponent of \(3\) is \(1\) (i.e., \(3^1\)).
Thus, the GCF is: \[ GCF = 3^1 = 3 \]
Therefore, the greatest common factor of 75, 66, 63, and 48 is \(\boxed{3}\).