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Friday, October 12, 2012

Greatest Common Factor, GCF

Category: Arithmetic

"Published in Newark, California, USA"

Given the following numbers: 252, 240, 288, and 204. What is their Greatest Common Factor (GCF)?

Solution:

There are two ways in getting their GCF. I will show the both methods and let's see which method is your preference.

Method 1: You can use the intersection method. You have to  get their factors of each given numbers. 

A = 252 = (1 x 252), (2 x 126), (3 x 84), (4 x 63), (6 x 42), (7 x 36), (9 x 28), (12 x 21), (14 x 18)
A = (1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252)

B = 240 = (1 x 240), (2 x 120), (3 x 80), (4 x 60), (5 x 48), (6 x 40), (8 x 30), (10 x 24), (12 x 20), (16 x 15)
B = (1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240)

C = 288 = (1 x 288), (2 x 144), (3 x 96), (4 x 72), (6 x 48), (8 x 36), (9 x 32), (12 x 24), (16 x 18)
C = (1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 72, 96, 144, 288)

D = 204 = (1 x 204), (2 x 102), (3 x 68), (4 x 51), (6 x 34), (12 x 17)
D = (1, 2, 3, 4, 6, 12, 17, 34, 51, 68, 102, 204)

Rewrite the factors of each given numbers,

A = (1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252)
B = (1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240)
C = (1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 72, 96, 144, 288)
D = (1, 2, 3, 4, 6, 12, 17, 34, 51, 68, 102, 204)

If A ∩ B ∩ C ∩ D = (1, 2, 3, 4, 6, 12)


Therefore, their GCF is 12.


Method 2: You can use the continuous division method. You have to think their factors as much as you can while dividing the numbers until they can't divide anymore. 

             2          252       240       288      204
                  2   │  126       120       144      102
                     3    63         60         72        51
                             21         20         24        17

At this stage, 17 is already a prime number. Therefore, the GCF is 2 x 2 x 3 = 12

Note: Please remember the procedure in getting the GCF of the numbers very well because you will use this method later in simplifying a fraction into a lowest term. Also, you must memorize or remember the prime and composite numbers all the time.