## Kiss those Math Headaches GOODBYE! ### How to Find the GCF — FAST!

Time-saving tips are great, right? So I’d like to share a time-saving tip for math.

This tip SIMPLIFIES the process of finding the greatest common factor (GCF) for two numbers, a good thing to know when simplifying fractions, reducing proportions, etc.

First, have you ever noticed that when students search for a GCF, they sometimes don’t know when to stop searching? This tip alleviates that problem, for it tell students exactly when they can stop testing numbers.

It turns out that students can stop testing when they reach the DIFFERENCE between the two numbers whose GCF they’re trying to find.

As an easy example, let’s say you need to find the GCF for 16 and 20.

All you do is subtract 16 from 20, to get the difference, 4, and this number — 4 — is the largest number that could POSSIBLY go into both 16 and 20 evenly.

Once you know that, just test 2, 3, and 4 to find the highest one that goes into 16 and 20. Of course that would be 4, so you got the GCF right off the bat, in this case.

Keep in mind that that greatest possible greatest common factor is not necessarily the true, greatest common factor. But it does set an upper limit for GCFs, and having that upper limit really reduces kids’ stress.

Another example:  find the GCF for 25 and 35.

35 – 25 = 10, so 10 is the greatest possible GCF. But of course 10 does not go into 25 and 35, so 10 is not the GCF. Check the numbers less than 10, and you’ll see that 5 is the GCF. But no more checking above 10, as kids are likely to do, unless you tell them when to stop.

I have dubbed this mathematical object the GPGCF, for Greatest Possible Greatest Common Factor, and I’ve found that students really appreciate learning it’s there — to alert them when it’s “quitting time.”

Try it out yourself, whenever it next flows with your lesson. Let me know what kind of reaction you get from the kids, and good luck.

By the way, if you’d like to explain to your students why this trick works, here’s a way to look at it. If you think about this situation via the number line, the GPGCF is simply the distance between the two numbers whose GCF you’re trying to find. Let’s go back to our first example: searching for the GCF for 16 and 20. The difference between 20 and 16, 4, is the distance between 16 and 20 on the number line. So if any number does go into both 16 and 20, it cannot be larger than 4, since that’s the space between the numbers.

To see this clearly, imagine that you wonder for a moment if 8 might be the GCF for 16 and 20. Well it is true that 8 does go into the first of these numbers, 16. But the next number that 8 goes into evenly must be 8 greater than 16, or 24. In other words, 8 is going to “leap past” 20, by hitting 24, when it goes into its next multiple. So the space between the numbers — 4 in this case — gives you the biggest number that could possibly fit into both numbers.

DIRECTIONSs:  Given each pair of numbers, first find the GPGCF. Then use the GPGCF to help you find the GCF.

a)  6, 10

b)  8, 12

c)  12, 15

d)  12, 20

e)  14, 28

f)  18, 26

g)  27, 36

h)  36, 48

i)  42, 60

j)  72, 80

a)  6, 10   GPGCF = 4   GCF =  2

b)  8, 12   GPGCF = 4   GCF =  4

c)  12, 15   GPGCF = 3   GCF =  3

d)  12, 20   GPGCF = 8   GCF =  4

e)  14, 28   GPGCF = 14   GCF =  14

f)  18, 26   GPGCF = 8   GCF =  2

g)  27, 36   GPGCF = 9   GCF =  9

h)  36, 48   GPGCF = 12   GCF =  12

i)  42, 60   GPGCF = 18   GCF =  6

j)  72, 80   GPGCF = 8   GCF =  8

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! #### Comments on: "How to Find the GCF — FAST!" (14)

1. Alex said:

THANKS SOO MUCH, i hate trying to find the gcf for huge numbers, this really helped :p

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2. praveen said:

i liked it …..
but for bigger numbers it really becomes tough…
ex. for 42 and 60 ..

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3. Jo said:

Brilliant! Math stresses me out… this helped… a lot! Thank you.

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4. continuingtobloom said:

Thank you. As an older adult returning to school I am facing down my old nemesis Algebra. I can use all the tips and tricks I can find!

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5. Evan said:

This helped a lot one of the best tricks i’v read.

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6. manisha said:

can u say …..how its done for three numbers

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• Hi Manisha,
Thanks for your note. Yes, I have a whole video on how to find the GCF for three or more numbers, using an original technique that’s quite a time-saver, once you get the hang of it. You’ll find it here:

There’s also a blogpost on finding the GCF for three or more numbers, which uses the more standard technique, and you’ll find that here:
https://mathchat.me/2015/03/26/how-to-find-the-gcf-of-3-numbers-fast-no-prime-factorizing/

Hope that helps. Let me know.
— Josh

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• Wendy said:

WOW …I cant wait to share this with my kids and grand kids…This is amazing tip. Why do they not teach this to kids in school???

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7. Rick said:

So the GPGCF of 49-22 is 27? That doesn’t seem right…

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• Hi Rick. I see why you’re saying that, because 27 is larger than 22. In such a case, the GPGCF is the lower of the two numbers, 22. The lower of the two numbers presents another upper bound on the GCF. So you can use the factors of 22, which are: 22, 11, and 2, and see that none of them goes into 49. That means the GCF of 22 and 49 is simply 1. You raise a good point, Rick, and I address it in one of my videos on the GCF. There are two possible GPGCFs: the difference between the numbers and also the lower number. Choose whichever is smaller as the true GPGCF.

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8. Tabby said:

What about 225 and 120? 105 doesn’t seem right…

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• I believe that you did not read my article
carefully enough. I did not say that the difference
between two numbers is the GCF.

I said that it is the greatest possible GCF.

So for 225 and 120, 105 is the greatest possible GCF,
the GPGCF, but that doesn’t mean that 105 is the GCF.

If you read the article again, you’ll see that what it
says it that once you find the GPGCF, you next find the
factors of the GPGCF, and then you find the largest of those
factors that does goe into the two numbers under consideration.

So for 225 and 120, you look at the factors of 105,
which are (greatest to least): 35, 21, 15, 7, 5, 3.

And it turns out that of those six numbers, the largest that
goes into both 225 and 120 is 15, so for these two numbers,
the GCF is 15.

If you re-read the post, you will see that this is what it
says.

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9. I was working in Java to change this into code and 90 and 15 won’t seem working. and I have read your article great job.

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• Hi Daniel, The trick works just fine with any pair of numbers. For 90 and 15, the GPGCF = 75 since 90 – 15 = 75. Obviously 75 doesn’t divide evenly into 15, since 75 > 15. So skip 75 and start going through the factors of 75 from greatest to least. That list of factors: 25, 15, 5, 3. Note that 25 also can’t divide into 15, since 25 > 15. Next number on the list is 15. Note that 15 does divide evenly into both 15 and 90. So 15 is the GCF. I don’t know why this is not working for you in your Java program (I’m not a programmer). But here’s one other thing that could streamline your program: for every pair of numbers there’s a second, rather obvious limiting number. The smaller of the two original numbers is a limiting number since no number larger than x can divide into x. So in this problem, 15 is a limiting number since no number > 15 can divide into 15. So in this problem, you can start out by checking 15 and its factors. You’ll immediately find that 15 does divide into 90, so you can get the GCF = 15 that way as well. I hope this makes sense to you. Let me know if you have any other questions, and good luck with the programming. Sounds interesting! — Josh

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