Download Notes on Greatest Common Factor

Notes on Greatest Common Factor
Name_________________________
To begin, let's review how to divide monomials. In order to divide two monomials, one should
divide/simplify the coefficients and subtract the exponents of like variables.
Simplify.
1. 24 x8  4 x 2
2.
56 y
7y
3.
5a 20bz 3
3a 4 z 2
--------------------------------------------------------------------------------------------------------------------The greatest common factor, or GCF, of two or more integers (...,-2, -1, 0, 1, 2, ...) is the
largest natural number (1, 2, 3, ...) that will divide evenly into all of the integers a natural number
of times.
For example, what is the greatest common factor of 10 and 25?
What is the GCF of 12, 36, and 48?
--------------------------------------------------------------------------------------------------------------------Find the greatest common factor of the given numbers.
4. 16 and 18
5. 24, 36, and 56
6. 27 and 80
--------------------------------------------------------------------------------------------------------------------It is important to note the word common embedded in the term greatest common factor. This
implies, of course, that the two numbers have something in common. This concept can best be
seen by making factor trees. Make factor trees for the two numbers in number four above.
It would probably not be an efficient use of time to make factor trees for every GCF problem.
Students should use their knowledge of multiplication and division, as well as the calculator, in
order to complete the problems in a timely fashion.
--------------------------------------------------------------------------------------------------------------------
However, suppose the factor tree idea is extended to variables. In order to find the greatest
common factor of the expressions below, it may help to make a factor tree or write the quantities
in expanded form (ex. x 4  x  x  x  x ).
7. x 2 and x 5
8. y 4 and y 3
9. z and z10
10. a 3 and b6
--------------------------------------------------------------------------------------------------------------------Let's summarize the findings from 7-10.
The greatest common factor of two or more like variable quantities is the variable raised to
the lowest exponent in all of the quantities.
70
For example, the GCF of c88 , c91 , and c 70 is c .
--------------------------------------------------------------------------------------------------------------------Find the greatest common factor of the quantities below.
11. d 100 and d 56
12. k 7 , k 5 , and k 11
13. p 4 and q 4
--------------------------------------------------------------------------------------------------------------------In order to find the greatest common factor of two or more monomials, simply find the greatest
common factor of the coefficients, and then find the GCF of the variable parts.
For example, what is the GCF of 6x 4 y 7 and 8 y 3 ?
--------------------------------------------------------------------------------------------------------------------Find the greatest common factor of the quantities below.
14.
5x and 30x 2
15. 10a 5b and 4a 4
16. 27 y 6 , 18y 3 , and 45y 2
--------------------------------------------------------------------------------------------------------------------In closing, the two skills discussed today (dividing monomials and finding the greatest common
factor) are merely building blocks to a more complicated concept to be discussed tomorrow.
Without mastery of these two skills, students will not succeed at the concept taught tomorrow.
Homework on Greatest Common Factor
Name_________________________
Divide. You are NOT finding the greatest common factor. Assume all variable represent
nonzero quantities.
1. 14 x 4  2 x
2. 64 y10  4 y 2
3.
5z 2
5z 2
19c18
3c5
4.
24a 8
2
5. b88  b 22
6.
7.
50k 7
10k 7
8. m  m
9.
42 p 4 q 5
14q
--------------------------------------------------------------------------------------------------------------------Find the greatest common factor of the given quantities. If the greatest common factor is '1',
write PRIME.
10. 13 and 65
11. 72 and 35
12. 96, 48, and 56
13. 9, 36, and 126
14. r and r 3
15. s10 and s8
16. t 12 , t 7 , and t 21
17. u 5 and w30
18. xy 4 and y 5
19. a11b 5 and a10b10
20. c12 d 21e8 , c9 de8 , and c12 d 2 e8
21. 6k and 20k
22. 30m3 and 15m 2
23. 6x3 , 4x 2 , and 2x
24. y 5 , 7 y 4 , and 21y3
25. 18a 2b 2 and 40a 2b 2
26. 12x9 , 5 y 4 , and 7z
27. 22cd and 33d