SOLUTION: I'm trying to help my son, 6th grade, with his home work. Could use any help you have to offer. When multiplying the greatest common factor (GCF) and the least common multiple (L

Question 17726: I'm trying to help my son, 6th grade, with his home work. Could use any help you have to offer.
When multiplying the greatest common factor (GCF) and the least common multiple (LCM) of two numbers the product is always equal to the product of the original two numbers. Why?
Is there an easy to say theorem or "rule" that you could share.
What we came up with: The least commom multiple and the greatest common factor are common to the two numbers. Because they are the smallest and the largest common numbers, when they are multiplied their product is equal to the product of the original two numbers.
Answer by ivy12003(22) (Show Source):
You can put this solution on YOUR website!
Thinking about this question logically, the greatest common factor of a number is always equal to the number itself, and the least common factor of a number is always equal to 1. For example, the greatest common factor of 12 is 12 (the number itself) and the least common factor is always 1. (12 * 1 = 12)

Building on that information, if the GCF and the LCF of two numbers will always equal to the product of the two numbers because esentially, you are mulitplying them together and mulitplying them by 1 twice.

Example:

10 and 12

LCF: 1, 1

GCF: 10, 12

Equation: 1 * 1 * 10 * 12 (you don't necessarily have to multiply by one twice because you'll still get the same product)

1 * 1 * 10 * 12 = 120

10 * 12 = 120

In other words, you are mulitplying the two numbers together when you are mulitplying the GCF and the LCF but in a longer equation.

Hope this information is helpful, please do not hesitate to ask if you are confused!