## Kiss those Math Headaches GOODBYE!

### Math Trick: Find the GCF for 3 or More Numbers

Find the GCF, your teacher says … not just for 2 numbers, but for 5 of them.

And yes, you need to do it by prime factorizing.

Can’t you just hear the students’ groans?!

But what if there were a way to do this without prime factorizing? Could it really be?

Yes!

What I’m about to teach you is a technique that lets you find the GCF of as many numbers as you wish, and with much greater ease than the old factoring technique. (by the way, I don’t really hate the factoring technique … it actually teaches you a lot about numbers … but it can get annoying!).

So why don’t they teach this new way in school? No idea. But let’s just focus on the technique because once you do, you’ll be so much faster at finding the GCF …  you’ll be amazing your friends and your teacher, too!

So just kick back, watch the video — and learn …. then do the practice problems at the end of the video, to become a whiz! And remember, if you ever want extra help in the form of tutoring, I’m available — worldwide — thanks to the power of online videoconferencing.

Enjoy!

— Josh

### Fun Math Trick: Find the GCF for Any Two Numbers

“Get the GCF for these two numbers,” your teacher says.

“Easy,” your teacher replies. “Do what I told you yesterday.”

But perhaps you don’t remember. Or maybe you don’t like the technique your teacher used, telling you to prime factorize the numbers …

Even though that felt awful, now you’re in luck because I’m going to show you another way to find the GCF, one that’s more intuitive and easier than the prime factoring technique. So just sit back and watch the following video. Then do the practice problems at the end of the video. And you’ll know something that even your teacher probably doesn’t know … a thoroughly original way to find the GCF of two numbers.

### How to Simplify Fractions — FAST

So c’mon … everything that can be said about simplifying fractions has been said … right?

Not quite! Here’s something that might just be original … a hack to smack those fractions down to size.

Suppose you’re staring at an annoying-looking fraction:  96/104, and it’s annoying the heck out of you, particularly because it’s smirking at you!

But it won’t smirk for long. For you open up your bag of hacks (obtained @ mathchat.me) and …

1st)  Subtract to get the difference between numerator and denominator. I also like to call this the gap between the numbers. Difference (aka, gap) = 104 – 96 = 8.

NOTE: Turns out that this gap, 8, is the upper limit for any numbers that can possibly go into BOTH 96 and 104. No number larger than 8 can go into both. And this is a … HACK FACT:  The gap represents the largest number that could possibly go into BOTH numerator and denominator. In other words, the gap is the largest possible greatest common factor (GCF).

2nd)  Try 8. Does 8 go into both 96 and 104? Turns out it does, so smack the numerator and denominator down to size:  96 ÷ 8 = 12, and 104 ÷ 8 = 13.

3rd)  State the answer:  96/104 = 12/13.

Is it still smirking? I think … NOT!

Try another. Say you’re now puzzling over:  74/80.

1st)  Subtract to get the gap. 80 – 74 = 6. So 6 is the largest number that can possibly go into BOTH 74 and 80.

2nd)  So try 6. Does it go into both 74 and 80? No, in fact it goes into neither number.

NOTE:  Turns out that even though 6 does NOT go into 74 OR 80, the fact that the gap is 6 still says something. It tells us that the only numbers that can possibly go into both 74 and 80 are the factors of 6:  6, 3 and 2. This, it turns out, is another … HACK FACT:  Once you know the gap, the only numbers that can possibly go into the two numbers that make the gap are either the factors of the gap, or the gap number itself.

3rd)  So now, try the next largest factor of 6, which just happens to be 3. Does 3 go into both 74 and 80? No. Like 6, 3 goes into neither 74 nor 80. But that’s actually a good thing because now there’s only one last factor to test, 2. Does 2 go into both 74 and 80? Yes! At last you’ve found a number that goes into both numerator and denominator.

4th)  Hack the numbers down to size:  74 ÷ 2 = 37, and 80 ÷ 2 = 40.

5th)  State the answer. 74/80 gets hacked down to 37/40, and that fraction, my dear friends, is the answer. 37/40 the final, simplified form of 74/80.

O.K., are you ready to smack some of those fractions down to size? I believe you are. So here are some problems that will let you test out your new hack.

As you slash these numbers down, remember this rule. In some of these problems the gap number itself is the number that divides into numerator and denominator. But in other problems, it’s not the gap number itself, but rather a factor of the gap number that slashes both numbers down to size. So if the gap number itself doesn’t work, don’t forget to check out its factors.

Ready then? Here you go … For each problem, state the gap and find the largest number that goes into both numerator and denominator. Then write the simplified version of the fraction.

a)   46/54
b)   42/51
c)   48/60
d)   45/51
e)   63/77

a)   46/54:  gap = 8. Largest common factor (GCF) = 2. Simplified form = 23/27
b)   42/51:  gap = 9. Largest common factor (GCF) = 3. Simplified form = 14/17
c)   48/60:  gap = 12. Largest common factor (GCF) = 12. Simplified form = 4/5
d)   45/51:  gap = 6. Largest common factor (GCF) = 3. Simplified form = 15/17
e)   63/77:  gap = 14. Largest common factor (GCF) = 7. Simplified form = 9/11

Josh Rappaport is the author of five math books, including the wildly popular Algebra Survival Guide and its trusty sidekick, the Algebra Survival Workbook. Josh has been tutoring math for more years than he can count — even though he’s pretty good at counting after all that tutoring — and he now tutors students in math, nationwide, by Skype. Josh and his remarkably helpful wife, Kathy, use Skype to tutor students in the U.S. and Canada, preparing them for the “semi-evil” ACT and SAT college entrance tests. If you’d be interested in seeing your ACT or SAT scores rise dramatically, shoot an email to Josh, addressing it to: josh@SingingTurtle.com  We’ll keep an eye out for your email, and our tutoring light will always be ON.

### 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!

### 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

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.