Greatest Common Factors

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 
and 
54 = 2 * 3 * 3 * 3

        36 = 2 * 2 * 3 * 3

        54 = 2 * 3 * 3 * 3

    Both have 2 * 3 * 3 in common:
    GCF =  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 next-largest 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




PRACTICE:     ("Click" on Check answer or use the answer key below.)

1) What is the greatest common factor (G.C.F.) of 12 and 10.

  A) 1 B) 2 C) 4 D) 5 E) 6
 


2)   What is the greatest common factor (G.C.F.) of 20 and 30?

  A) 1 B) 2 C) 4 D) 5 E) 10
 


3)   What is the greatest common factor of 8 and 40?

  A) 1 B) 2 C) 4 D) 8 E) 10
 


4)   4) What is the greatest common factor of 27 and 12?

  A) 1 B) 2 C) 3 D) 6 E) 9
 


5)   Find the greatest common factor (G.C.F.) of 8 and 27.

  A) 1 B) 2 C) 3 D) 4 E) 8
 


6)   Find the greatest common factor (G.C.F.) of 40 and 60.

  A) 1 B) 2 C) 4 D) 10 E) 20
 


7)   Find the greatest common factor (G.C.F.) of 48 and 36.

  A) 1 B) 4 C) 6 D) 12 E) 18
 


8)   Find the greatest common divisor (G.C.F.) of 18 and 8.

  A) 1 B) 2 C) 4 D) 6 E) 8
 


ANSWERS: 1 -B     2 -E     3 -D     4 -C     5 -A     6 -E     7 -D     8 -B





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