Identify the factors of 9X³Y²
= __________
Answers
Answer:
Greatest Common Factor and Least Common Multiple
GCF and LCM
Motivation Problem for GCF
How are factors involved in the following scenario?
Gary has 20 table tennis balls and 16 paddles. He wants to sell common sized packages containing both paddles and balls. What is the greatest number of packages he can sell with no left over balls or paddles?
We use factors to search for solutions to the first problem.
1 · 20 = 20 and 1 · 16 = 16
One package with 20 balls and 16 paddles.
2 · 10 = 20 and 2 · 8 = 16
Two packages each with 10 balls and 8 paddles.
4 · 5 = 20 and 4 · 4 = 16
Four packages each with 5 balls and 4 paddles.
The question can be answered by finding the greatest common factor. If Gary wants to divide the balls and paddles into packages with each package containing the same number of balls then we are looking for a number that is a factor of both. This is what we would call a common factor. The above illustration shows that Gary has three options for packaging the balls and paddles with one, two, or four packages. If we want to create as many packages as possible then we are looking for the greatest common factor.
Example:
Set of factors of 20 is {1, 2, 4 , 5, 10, 20}
Set of factors of 16 is {1, 2, 4 , 8, 16}
From these two lists we see that the greatest common factor of 20 and 16 is 4, so Gary would be able to sell four packages each containing four paddles and five balls.
This problem motivates a need for being able to determine the greatest common factor for two or more values.
Greatest Common Factor (GCF)
Definition of Greatest Common Factor (GCF) :
The greatest common factor (GCF) of two natural numbers n and m is the greatest natural number k that is a factor of both n and m. We symbolize this as GCF(n, m) = k.
Example:
Relating this definition to the problem we solved on the previous page, we see that the two numbers n and m are the numbers 20 and 16. We may write the solution symbolically as GCF(20, 16) = 4. This says "the greatest common factor of 20 and 16 is 4."
The number k in the definition corresponds to the 4 in the problem. It is the factor of the greatest value shared by both n and m. The number k is the Greatest Common Factor of 20 and 16.
One method for finding the greatest common factor is to choose the value that is the greatest value in the intersection of the sets of common factors.
Example: Find the GCF(20, 16).
The set of factors of 20 is {1, 2, 4, 5, 10, 20} and the set of factors of 16 is {1, 2, 4, 8, 16}.
{1, 2, 4, 5, 10, 20} ∩ {1, 2, 4, 8, 16} = {1, 2, 4} which has greatest value 4.
So, GCF(20, 16) = 4.
Notice that an intermediate step in this method applied to the problem at the beginning of the session gives all possibilities for packaging all the balls and paddles: one, two, or four packages.
Example: Find GCF(18, 24).
The set of factors of 18 is {1, 2, 3, 6, 9, 18}
and set of factors of 24 is {1, 2, 3, 4, 6, 8, 12, 24}.
{1, 2, 3, 6, 9, 18} ∩ {1, 2, 3, 4, 6, 8, 12, 24} = {1, 2, 3, 6} which has greatest value 6.
The GCF (18, 24) = 6.
Step-by-step explanation:
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