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Posts tagged ‘Find the GCF’

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

 

 

 

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Find the GCF for 2 Numbers


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

How, you ask.

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

But let’s say you don’t remember. Or let’s say you don’t like the technique your teacher used, telling you to prime factorize the numbers … well then you’re in luck because I’m going to show you another way to find the GCF, one that is 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 Find the LCM for 3+ Numbers — FAST!


Is there a quick-and-easy way to find the LCM for three or more numbers … WITHOUT prime factorizing?

Of course! We’ll demonstrate the technique by finding the LCM for 10, 14, 20.

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To begin, use the technique for finding the GCF for 10, 14, 20 that’s shown in my post:  How to Find GCF for 3+ Numbers — FAST … no prime factorizing. If you don’t want to go to that post, no worries. I’ll re-show the technique here.

1st)   Write the numbers from left to right:

……….   10     14     20

[The periods: …… are just to indent the lines. They have no mathematical meaning.]

2nd)  If possible, rip out a factor common to all numbers. The factor 2 is common. So divide the three numbers by 2 [10 ÷ 2 = 5 and 20 ÷ 2 = 10] and show the result below:

2   |       10     14     20
……….   5      7      10

3rd)  At this point, notice there’s no number that goes into the remaining numbers: 5, 7, 10. That means you’ve found that the GCF is the number pulled out, 2. At this point we’re at a crossroads. We’re done finding the GCF, but now we’re at the start of a new process, finding the LCM.
To proceed toward getting the LCM, see if there’s any number that goes into any pair of remaining numbers. Well, 5 goes into 5 and 10. So divide both those numbers by 5 [5 ÷ 5 = 1 and 10 ÷ 5 = 2] , and show the results below:

2   |       10     14     20
5   |         5      7      10
………..  1       7       2

Notice that if there’s a number 5 doesn’t go into, you leave that number as is. So leave the 7 as 7.

4th)  Repeat. See if there’s a number that goes into two of the remaining numbers. Since nothing goes into 1, 7, and 2, we’re done. To get the LCM, multiply all of the outer numbers. That means you multiply the numbers you pulled out on the left (2 and 5), and also multiply the numbers at the bottom (1, 7 and 2). Ignoring the meaningless 1, you have:  2 x 5 x 7 x 2 = 140, and that’s the LCM.

To see the process in more depth, let’s find the LCM for … not three, not four … but five numbers:
6, 12, 18, 30, 36.

1st)   Write the numbers left to right:

………  6     12     18     30     36

2nd)  If possible, rip out a common factor.  2 is common, so divide all by 2 and show the results below:

2     |    6     12     18     30     36
………. 3      6       9     15     18

3rd)  Repeat. See if there’s a number that goes into the five remaining numbers. 3 goes into all, so divide all by 3 and show the results below:

2     |    6     12     18     30     36
3     |    3       6       9     15     18
……..   1       2       3       5       6

4th)  Repeat. See if any number goes into the last remaining numbers. Nothing goes into all of them, so now you get the GCF by multiplying the left-hand column numbers. GCF = 2 x 3 = 6.
Proceeding to find the LCM, look for any number that goes into two or more of the remaining numbers. One such number is 3, which goes into the remaining 3 and 6. Divide those numbers by 3 and leave the other numbers as they are.

2     |    6     12     18     30     36
3     |    3       6       9     15     18
3     |    1       2       3       5       6
……… 1       2       1       5       2

5th)  Interesting! Notice that 2 goes into the two remaining 2s, so pull out a 2 and show the results below:

2     |    6     12     18     30     36
3     |    3       6       9     15     18
3     |    1       2       3       5       6
2     |    1       2       1       5       2
……..   1      1        1       5       1

6th)  We’ve whittled the bottom row’s numbers so far down that finally there’s no number that goes into two or more of them (except 1, which doesn’t help). So we have all the numbers we need to find the LCM. Multiply them together. The left column gives us:  2 x 3 x 3 x 2. The bottom row gives us 1 x 1 x 1 x 5 x 1. Multiply all of those (non-1) numbers together, you get:
2 x 2 x 3 x 3 x 5 = 180, and that is the LCM! Pretty amazing, huh? And no prime factorizing, to boot.

Some people find that this process takes a bit of practice to get used to it. So here are a few problems to help you become an LCM-finding expert!

a)  12, 18, 30
b)   8, 18, 24
c)  15, 20, 30, 35
d)  16, 24, 40, 56
e)   16, 48, 64, 80, 112

And the answers. LCM for each set is:

a)   180
b)   72
c)   420
d)   1680
e)   6720

How to find the GCF of 3+ Numbers — FAST … no prime factorizing


Suppose you need to find the GCF of three or more numbers, and you’d really prefer to avoid prime factorizing. Is there a way? Sure there is … here’s how.

 

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Example:  Find the GCF for  18, 42 and 96

Step 1)  Write the numbers down from left to right, like this:

………. 18     42     96

[FYI, the periods: …. are there just to indent the numbers. They have no mathematical meaning.]

Step 2)  Find any number that goes into all three numbers. You don’t need to choose the largest such number. Suppose we use the number 2. Write that number to the left of the three numbers. Then divide all three numbers by 2 and write the results below the numbers like this:

2    |  18     42     96
……..  9     21     48

Step 3)  Find another number that goes into all three remaining numbers. It could be the same number. If it is, use that. If not, use any other number that goes into the remaining numbers. In this example, 3 goes into all of them. So write down the 3 to the left and once again show the results of dividing, like this:

2    |  18     42     96
3    |    9     21     48
……… 3      7      16

Step 4)  You’ll eventually reach a stage at which there’s no other number that goes into all of the remaining numbers. Once at that stage, just multiply the numbers in the far-left column, the numbers you pulled out. In this case, those are the numbers:  2 and 3. Just multiply those numbers together, and that’s the GCF. So in this example, the GCF is 2 x 3 = 6, and that’s all there is to it.

Now try this yourself by doing these problems. Answers are below.

a)   18, 45, 108
b)   48, 80, 112
c)   32, 72, 112
d)   24, 60, 84, 132
e)   28,  42, 70, 126, 154

Answers:
a)   GCF =  9
b)   GCF =  16
c)   GCF =  8
d)   GCF =  12
e)   GCF =  14

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!