As a teacher, some of your students have become con. fused over the concepts of greatest common factor GCF) and least common multiple (LCM). One student said: "If the factors of a number are less than the mul. tiples, why does the GCF ask for the greatest of the factors and the IM ask for the least of the multiples?
Explain how you would answer this question.
I would explain to the student that the concepts of greatest common factor (GCF) and least common multiple (LCM) focus on different aspects of numbers.
The GCF asks for the greatest common factor shared by two or more numbers. Factors are numbers that divide evenly into another number, meaning they are smaller than the original number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. The GCF seeks to identify the largest factor that is common to all given numbers. It helps us find the highest common factor that can divide into all of the numbers without leaving any remainder.
On the other hand, the LCM asks for the least common multiple of two or more numbers. A multiple is a number that can be obtained by multiplying a given number by another whole number, meaning they are larger than the original number. For example, the multiples of 12 are 12, 24, 36, 48, 60, and so on. The LCM aims to find the smallest multiple that is divisible by all the given numbers without leaving any remainder.
Thus, the GCF focuses on identifying the highest factor shared by the numbers, while the LCM focuses on finding the smallest multiple common to them. These concepts have different purposes and are used in various mathematical problems and scenarios.