GCF = Greatest Common Factor
This topic wants you to find the largest thing that will divide into each part of a problem.
Letâ€™s start with numbers.
Example 1:
Find the GCF of 10 and 15.
Solution:
A factor is a number that will divide into another number.
10 can be divided by: 1, 2, 5 and 10. These are all the FACTORS of 10.
15 can be divided by: 1, 3, 5 and 15. These are all the FACTORS of 15.
A common factor is a number that BOTH numbers have in common from the list of factors.
The common factors of 10 and 15 are: 1 and 5.
The greatest common factor is the largest number fro the common factor list.
The greatest common factor of 10 and 15 is 5 because it is the largest number.
Example 2: Find the GCF of 12, 16 and 24.
Solution:
The factors of 12 are: 1, 2, 3, 4, 6 and 12.
The factors of 16 are: 1, 2, 4, 8 and 16.
The factors of 24 are: 1, 2, 3, 4, 6, 8, 12 and 24.
The common factors are 1, 2, and 4.
The greatest common factor is 4 because it is the largest from the list.
Lets try variables (letters).
Problem: Find the GCF of x^{3}y^{5}, x^{5}y^{4} and x^{8}y.
Solution:
Rename the parts of the problem into the basic meaning:
x^{3}y^{5} = xxxyyyyy
x^{7}y^{4 }= xxxxxxxyyyy
x^{8}y = xxxxxxxy
xxx are the MOST factors of the letter x and y is the MOST factors of the letter y that these three have parts have in common: so the GCF is x^{3}y.
Problem: a^{3}b^{6}c^{3}, a^{5}b^{3}c^{7} and ab^{2}c^{4}.
Solution:
a^{3}b^{6}c^{3 }= aaabbbbbbccc
a^{5}b^{2}c^{7 }= aaaaabbccccccc
ab^{3}c^{4 }=abbbcccc
a is the MOST factor of the letter a, bb are the MOST factors of the letter b and ccc are the MOST factors of the letter c that the parts have in common: so the GCF is ab^{2}c^{3}.
Problem: Find the GCF of 6m^{2}n^{4}p^{3}, 9m^{2}n^{4}p^{3} and 12m^{4}n^{3}p^{4}
Solution:
Find the GCF of the coefficients:
Factors are:
6 = 1, 2 and 3
9 = 1, 3, 9
12 = 1, 2, 3, 4, 6, 12
Common factors are:
1 and 3
Greatest common factor is 3.
Now, work on the letters:
m^{2}n^{4}p^{3} = mmnnnnppp
m^{2}n^{4}p^{3} = mmnnnnppp
m^{4}n^{3}p^{4} = mmmmnnnpppp
mm are the MOST factors of the letter m, nnn are the MOST factors of the letter n and ppp are the MOST factors of the letter that the parts have in common: so the GCF is m^{2}n^{3}p^{3}.
Then write your FINAL answer for the problem:
The GCF is 3 m^{2}n^{3}p^{3}.
