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Finding the Greatest Common Factor
Definition: The Greatest Common Factor (GCF) is the largest
number/expression that `
divides into two or more expressions evenly .
For Example: For the numbers 18 and 27, 3 is a common factor, but 9 is
the greatest
common factor , since 9 is the largest number that divides into 18 and 27
evenly.
Finding the GCF: One approach to finding the GCF is looking at the
prime factors that
occurs the least (look for the smallest exponent ) in each of the numbers or
expressions
that are involved. For instance, in the previous example , 18 and 27, factor each
number
into its prime factors.
The least exponent on the 3 is two and on the 2 is zero (since 27 does not
have any
factors of 2) so the GCF is 3^{2} = 9.
Another example of finding the GCF of 90 and 120:
The least exponent of each factor is one so the GCF is 2●3●5 = 30.
Examples for Finding the GCF of Algebraic Expressions :
The same approach is used to find the GCF of algebraic expressions —factor
into prime
factors first .
Example: Find the GCF of 12x^{2}y^{3}w and 20xy^{2}.
Choose the least exponent for each factor. So the GCF is 2^{2}●x●y^{2}
(3, 5 or w did not occur
in both expressions so they are not part of the GCF).
Example: Find the GCF of 3x^{3} + 6x^{2} and 6x^{2} – 24
The GCF is 3(x + 2)
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