The Greatest Common Factor (G.C.F.) of two numbers is
the largest number that is a divisor of both. It is sometimes
called the Greatest Common Divisor. It can be used to simplify (or
reduce) fractions. Don't let the "greatest" in the name fool you 
the GCF is no larger than the smallest of the numbers. 

Hints: 

GREATEST
is biggest or largest. 

COMMON is
something shared or in common 

FACTORS
are the parts of multiplication facts. 


EXAMPLE: 

Find the Greatest Common Factor (G.C.F.) of 6 and 10. 



6 = 2 * 3 You can divide 6 by 2 or by 3 

6 = 1 * 6 You can divide 6 by 1 or by 6 

Therefore 1, 2, 3, and 6 are all factors of six. 



10 = 2 * 5 You can divide 10 by 2 or by 5 

10 = 1 * 10 You can divide 10 by 1 or by 10 

Therefore 1, 2, 5, and 10 are all factors of ten. 



Both 6 and 10 can be divided by 1 and by 2; 2 is greater
than 1, so 2 is the Greatest Common Factor (G.C.F.) of 6 and 10. 


You can also use the prime
factorization method to find the Greatest Common Factor: 
EXAMPLE: 

Find the Greatest Common Factor (G.C.F.) of 36 and 54. 



36 = 2*2*3*3 

54 = 2*3*3*3 



Both have 2*2*3 in common: 

G.C.F = 2*3*3 = 18 


Usually you can find the Greatest Common
Factor fairly easily by experimenting with possible divisors: 


Start with the smaller number; it is the largest divisor
of itself.
Is it a divisor of the larger number? If so, you have the G.C.F.;
if not:
What is the nextlargest divisor of the smaller number; is IT a
divisor of the other number?
Continue until you find a number that will divide into BOTH. Sometimes
only the number '1' will work as a common divisor; for example: 21 and
16 have no common factor other than 1. 


EXAMPLE: 

Find the Greatest Common Factor (G.C.F.) of 10 and 30. 



Will 10 go into 30? Yes: G.C.F. =
10 



Find the Greatest Common Factor (G.C.F.) of 24 and 18. 



Will 18 go into 24? No.
Next pair: 2 * 9 = 18
Will 9 go into 24? No.
Next pair: 3 * 6 = 18
Will 6 go into 24? Yes:
G.C.F. = 6 