Final answer:
The greatest common factor (GCF) of 50, 56, and 105 is 70.
Step-by-step explanation:
To find the greatest common factor (GCF) of 50, 56, and 105 using prime factorizations, we need to prime factorize each number.
The prime factorization of 50 is 2*5*5, which can be written as 2^1*5^2.
The prime factorization of 56 is 2*2*2*7, which can be written as 2^3*7^1.
The prime factorization of 105 is 3*5*7, which can be written as 3^1*5^1*7^1.
To find the GCF, we take the lowest power of each common prime factor. In this case, the common prime factors are 2 (power 1), 5 (power 1), and 7 (power 1). So, the GCF is 2^1*5^1*7^1, which simplifies to 2*5*7 = 70.