What is true about how we determine the greatest common factor (GCF)?

Determining the greatest common factor (GCF) involves identifying the largest positive integer that divides each of the numbers without leaving a remainder. This method ensures that you are finding the most significant commonality between the numbers in terms of their divisibility.

When considering the largest number that divides both numbers cleanly, you are effectively looking for a factor that they share, which is the essence of what the GCF represents. For example, if you were finding the GCF of 12 and 16, you would identify the factors of both numbers (12 has factors 1, 2, 3, 4, 6, 12; 16 has factors 1, 2, 4, 8, 16) and recognize that the largest factor they both share is 4. This focused method of locating shared divisibility directly ties in with the definition and process of finding the GCF.

In contrast, adding the factors or multiplying the numbers does not yield the GCF and serves entirely different mathematical purposes. Additionally, finding the least common multiple (LCM) pertains to a different operation, focusing instead on the smallest common multiple rather than the largest common factor.