Final answer:
The greatest common factor of 75, 8, and 21 is 1 because they do not share any common prime factors other than 1, indicating that they are coprime numbers.
Step-by-step explanation:
The question asks to find the greatest common factor (GCF) of the numbers 75, 8, and 21. The GCF is the largest number that divides each of the numbers without leaving a remainder. To find the GCF, we can use the method of prime factorization or utilize the Euclidean algorithm.
Since the numbers do not share many common factors (8 is even while 75 and 21 are odd), we can quickly determine that the GCF is 1, because 1 is the only number that divides all three given numbers.
To further elaborate, there are no other common factors because 8 is a power of 2, while 75 is divisible by 5 and 3, and 21 is divisible by 7 and 3. The lack of common prime factors among all the numbers implies the GCF is 1. This is also known as being relatively prime or coprime. Hence, the greatest common factor of 75, 8, and 21 is 1.