Assessment and Learning in Knowledge Spaces (ALEKS) Practice Exam

How does one find the greatest common factor (GCF) of two numbers?

1. By adding the two numbers

2. By determining the smallest number that divides both

3. By determining the largest number that divides both without a remainder

4. By multiplying both numbers together

The correct answer is: By determining the largest number that divides both without a remainder

Finding the greatest common factor (GCF) of two numbers involves identifying the largest number that can divide both of them without leaving a remainder. This method ensures that you are looking for the common divisors of the two numbers and selecting the highest of those divisors. To visualize this, consider two numbers, for example, 12 and 15. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 15 are 1, 3, 5, and 15. The common factors between the two sets are 1 and 3, with 3 being the greatest. This approach is essential as it provides a systematic way to find the largest shared divisor which plays a significant role in simplifying fractions, finding least common multiples, and various applications in numbers theory. In contrast, adding the two numbers doesn't provide any information about their common divisors, and multiplying them together gives you a product that does not denote any commonality in factors. The concept of the smallest number that divides both may lead to misinterpretation; instead, it's the largest number that meets the criterion that defines the GCF.