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Posts tagged ‘GPGCF’

How to Find the GCF for Three or More Numbers


To find the GCF for three or more numbers,  follow these steps:

1)  Determine which of the given numbers is smallest, then find the smallest difference between any pair of numbers.

2)  See what is smaller:  the smallest number, or the smallest difference. Whichever one  is smallest, that number is the GPGCF (Greatest Possible GCF). That means that this is the biggest number that the GCF could possibly be. Or, more formally we would say:  The GCF, if it exists, must be less than or equal to the GPGCF.

3)  Check if the GPGCF itself goes into all of the given numbers. If so, then it is the GCF. If not, list the factors of the GPGCF from  largest to the smallest and test them until you find the largest one that does divide evenly into the given numbers. The first factor (i.e., the largest factor) that divides evenly into the given numbers is, by definition, the GCF.

EXAMPLE:

Problem:  Find the GCF for 18, 30,  54.

1)  Note that the smallest number is 18, and  the smallest difference between the pairs is 12 [54 – 30 = 24;  54 – 18 = 36;  30 – 18 = 12] .

2)  Of those four quantities (the smallest number and the three differences), 12 is the least. This means that the
GPGCF = 12.

3) Check if 12 divides evenly into the three given numbers: 18, 30 and 54. In fact, 12 doesn’t divide evenly into ANY of these  numbers. Next we check the factors of 12, in order from largest to smallest. Those factors are: 6, 4, 3 and 2. The first of those that divides evenly into all three numbers is 6. [18 ÷ 6 = 3;  30 ÷ 6 = 5;  54 ÷ 6 = 9]. So the GCF = 6. And we are done.
MORE CHALLENGING PROBLEM:

Find the GCF for 24, 148, 200.

1)  Note that the smallest number is 24, and that the smallest difference between the pairs is 52 [200 – 148 = 52;  200 – 24 = 176;  148 – 24 = 124] .

2)  Of those four quantities (the smallest number and the three differences), 24 is the least. This means that for this problem, the GPGCF = 24.

3) Check if 24 divides evenly into the three given numbers: 24, 148 and 200. While 24 does divide evenly into 24, it does not divide evenly into 148 or 200. So next we check the factors of 24, in order from largest to smallest. Those factors are: 12, 8, 6, 4, 3 and 2. The first of those that divides evenly into the three given numbers is 4. [24 ÷ 4 = 6;  148 ÷ 4 = 37;  200 ÷ 4 = 50]. So the GCF = 4. And, once again, we are done.

The process may seem a bit long, but once you get used to it and start doing it in your mind, not on paper, you should find that it actually is quite fast. And you’ll find yourself figuring out the GCF for three or more numbers all in your mind — with no need for pencil and paper — while everyone around you will be making prime factor trees or using calculators. And surely that is a good feeling.

Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly  like the Algebra Survival Guide and companion Workbook, both of which are available on Amazon.com  Just click the links in the sidebar for more information! 

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From GPGCF to GCF … in two easy steps


Once you know the GPGCF, you’re two easy steps from finding the GCF.
[If you don’t know how, see my last post:  “Recent Insight on the GCF (and GPGCF)”] That’s one of the benefits of finding the GPGCF — speed in getting the GCF! Here are the short, sweet steps:

1)  Find all factors of the GPGCF, and list them from largest to smallest.

2) Starting with the largest factor and working your way down the list, test to find the first factor that goes into both numbers. The first (largest) to do so is the GCF. You can bet on it!

Example 1 (Easy):  Find GCF for 30 and 42.

1st)  GPGCF is 12. Factors of 12, greatest to least, are 12, 6, 4, 3 and 2.

2nd)  Largest factor to go evenly into 30 and 42 is 6. So 6 is GCF.

Example 2 (Harder):  Find GCF for 72 and 120.

1st)  GPGCF is 48. Factors of 48, greatest to least, are 48, 24, 16, 12, 8, 6, 4, 3 and 2.

2nd)  Largest factor to go evenly into 72 and 120 is 24. So 24 is GCF.

NOW TRY THESE —

For each pair:

1) Find GPGCF and say if it is the difference or smaller #.
2) List factors of GPGCF, greatest to least.
3) Find GCF.

a)  8 and 12

b)  16 and 40

c)  18 and 63

d)   56 and 140

ANSWERS:

a)  8 and 12
GPGCF = 4 (difference)
Factors of 4:  4 and 2
GCF = 4

b)  16 and 40
GPGCF = 16 (smaller #)
Factors of 16:  16, 8, 4 and 2
GCF = 8

c)  18 and 66
GPGCF = 18 (smaller #)
Factors of 18:  18, 9, 6, 3 and 2
GCF = 6

d)  56 and 76
GPGCF  =  20 (difference)
Factors of 20:  20, 10, 5, 4 and 2
GCF  =  4

Recent insight on the GCF (and GPGCF)


A while back I wrote a post about the GCF, and mentioned that there’s a number  related to it — a number that I call the GPGCF. “GPGCF” stands for the “Greatest Possible Greatest Common Factor.”

In short, the GPGCF is a number that sets an upper limit for the size of the GCF. I’ve seen many students struggle when searching for the GCF, seeking hither and yon for it. I had a sense that students were checking numbers that were too large. That’s what led me to try to figure out what must the the upper limit for the GCF.

If you check out that post (10/25/10), you’ll see that, for any two numbers, I said that the difference between those numbers has to be the GPGCF.

And I was correct, to a degree.

But I recently realized that my little theory needs modifying.

For while the difference between any two numbers can be the upper limit for the GCF, that difference is not the only quantity that can set an upper limit for the GCF. There’s another quantity that plays a role.

That other quantity, I recently realized, is the size of the smaller of the two numbers.

Take the numbers 8 and 24, for example.

The difference between these two numbers is 16, so I would have said that 16 is the upper limit for the GCF. But there’s actually another quantity that limits the size of the GCF, and that quantity is 8. For since the GCF of 8 and 24 must by definition fit into both 8 and 24, it must fit into 8. And common sense tells us that there’s no number larger than 8 that can fit into 8! So the size of this number — the smaller of the two numbers — also sets an upper limit for the size of the GCF.

So my revised theory about the GPGCF is this:  when you need to find the GCF for any two numbers, look at two quantities:  1) the smaller of the two numbers, and 2) the difference between the two numbers. Both of these quantities constrains the size of the GPGCF. So therefore, whichever of these is smaller IS the GPGCF. Once you’ve found the GPGCF, that makes it easier to find the actual GCF.

I know this sounds very abstract, so let’s look at a few examples to see what I’m blabbering on about.

Example 1:  What’s the GPGCF for 6 and 16?

Smaller number is 6; difference is 10.
6 and 10 both limit the size of the GCF, but
6 is less than 10, so 6 is the GPGCF.

Example 2:  What’s the GPGCF for 8 and 12?

Smaller number is 8; difference is 4.
4 is less than 8, so 4 is the GPGCF.

Example 3:  What’s the GPGCF for 30 and 75?

Smaller number is 30; difference is 45.
30 is less than 45, so 30 is the GPGCF.

Example 4:  What’s the GPGCF for 28 and 42?

Smaller number is 28; difference is 14.
14 is less than 28, so 14 is the GPGCF.

Now, let’s go one step further. From here, how do we figure out the GCF? I’ve done a bit more thinking about this, too, and I’ll share those ideas in my next post.