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6th grade
Course: 6th grade > Unit 6
Lesson 11: Greatest common factorGreatest common factor explained
The greatest common divisor (GCD) and greatest common factor (GCF) are the same thing. To find the GCD/GCF of two numbers, list their factors, identify the common factors, and choose the largest one. For example, the GCD/GCF of 12 and 8 is 4. Numbers with a GCD/GCF of 1 are called relatively prime. Created by Sal Khan.
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What about bigger numbers like 118 and 204(91 votes)- For bigger numbers, you definitely want to use the Euclidean algorithm, which is an easier and faster way to find the answer. For example:
gcd( 118, 204 )
= gcd ( 118, 204 - 118 )
= gcd ( 118, 86 )
= gcd ( 118 - 86, 86 )
= gcd ( 32, 86 )
= gcd ( 32, 86 - 32 )
= gcd ( 32, 54 )
= gcd ( 32, 54 - 32 )
= gcd ( 32, 22 )
= gcd ( 10, 22 )
= gcd ( 10, 2 )
= 2
The simplest variant of the Euclidean algorithm is to keep subtracting the smaller number from the bigger number until you find a problem easy enough that you know the answer to it. And the answer to that easier problem is the same as the answer to your harder problem.(164 votes)
Does 0 have a GCD?(31 votes)
No, that concept is only used for non-zero integers and polynomials. GCD/GCF is also a property of two numbers not of only one.(43 votes)
I need to know how to do it with bigger number that's what I do in the exercise.(16 votes)
If you have to find the GCD of bigger numbers, the fastest way is factoring and comparing the factors: If one or both numbers are prime, then your job is very fast.
Let's say you have 318 and 492
Start dividing by the lowest possible prime numbers like 2 and 3 and 5
318(2
159(3
53 --prime
so the factors of 318 are2353
492(2
246(2
123(3
41 -- prime
so the factors are22341
Line up the factors235322341
both have23
so the greatest common divisor of 492 and 318 will be2 times 3or 6
A shortcut is to refer to a table of factors and primes which will often give you the results of big numbers as
928 = 2⁵∙29
1189 = 29∙41
You can quickly see that the common factor is 29
so the GCD(928,1189) = 29(22 votes)
ls there any numer that has the factors 1 2 3 4 5 6 7 8 and 9(12 votes)
There can be more, of course, if you multiply 2/3/4/5/6/7/8/9 to 362880.(8 votes)
Is GCM a concept in math? I don't know if my teacher said that accidentally instead of GCF.(11 votes)
There shouldn't be "GCM" in math because multiples for values can go on and on forever; all you have to do is keep multiplying the numbers you have by common values.
However, there is certainly the concept and use of GCFs. They are the greatest common factor that divides two numbers, and one use is to simplify fractions. There are also "LCMs" (Least common multiples), and when you add or subtract fractions, you can find an LCM for a smaller value (instead of having to multiply everything together and get very large products for your numerator and denominator).
[R](5 votes)
- I don't get it. What is the difference between GCD and GCF?(8 votes)
They are the same(6 votes)
I'm not really shore what is the difference of (GCD) and (GCF)?(9 votes)- There isn't much of a difference. GCF, which stands for "Greatest common factor", is the largest value of the values you have, that multiplied by whole number is able to "step onto both".
For example, the GCF of 27 and 30 is 3, since if you add 3 repeatedly, it will equal 27 after it is added 9 times and equal 30 after adding 3 10 times.
On the other hand, 15 is not a common factor because though 15+15=30, 15 "skips over" 27. 9 is not a common factor because while adding 9 three times will equal 27, 9 will "skip over" 30 (jump from 27 to 36).
GCD stands for "Greatest common denominator". This is used when you are working with fractions and want to simplify them and find a common denominator so you can add and/or subtract them.(6 votes)
- Don't you have to multiply the two common factors to get the greatest? Does that rule only work for some?(4 votes)
- Is it possible the GCF does not always show up when using factorisation? I've encountered this a couple of times on the practice exercise. EG. GCD(80,32) with factorisation: 80= 8x10=2x4x2x5=2x2x2x2x5 or 2x40=2x2x20=2x2x10=2x2x5) and 32= 2x16=2x2x8=2x2x4=2x2x2. the gcd= 16 when you write out the list for both so why does 16 not show up when factorising 80? Does this make factorising unreliable in this case? It also happened when I factorised (40, 72) the gcd of 8 didn't show up when factorsing 40. 4x10=2x2x2x5. (no 8 there.) 72=8x9=2x4x3x3=2x2x2x3x3. Only one 8 is seen to be common here (8x9) so I answered GCD=4 (4x10 & 2x4) which was incorrect. The gcd did show up when I factorised (21, 70)=gcd 7. 21=3x7 and 70=7x10. GCD=7.Have I made an error in my arithmetic or is there something else going on here? Cheers, Sal!(3 votes)
When figuring out the GCD you need to first factor out your 2 numbers then select all the like factors and multiply them together. This includes multiple copies of the same factor, so for your example of GCD(80,32).
80 = 2 * 2 * 2 * 2 * 5 = 2^4 * 5 (the carrot,^, represents an exponent)
32 = 2 * 2 * 2 * 2 * 2 = 2^5
So when I look for common factors, I see that each of them have 4 - 2s, 2^4, as common factors and 2^4 = 2 * 2 * 2 * 2 = 16 so the GCD(80, 32) = 16(3 votes)
- Is there a video for LCM?(3 votes)
Video transcript
Welcome to the greatest
common divisor or greatest common factor video. So just to be clear, first of
all, when someone asks you whether what's the greatest
common divisor of 12 and 8? Or they ask you what's
the greatest common factor of 12 and 8? That's a c right
there for common. I don't know why it
came out like that. They're asking you
the same thing. I mean, really a divisor is
just a number that can divide into something, and a factor--
well, I think, that's also a number that can divide
into something. So a divisor and a factor
are kind of the same thing. So with that out of the way,
let's figure out, what is the greatest common divisor or
the greatest common factor of 12 and 8? Well, what we do is, it's
pretty straightforward. First we just figure out the
factors of each of the numbers. So first let's write all of the
factors out of the number 12. Well, 1 is a factor,
2 goes into 12. 3 goes into 12. 4 goes into 12. 5 does not to go into 12. 6 goes into 12
because 2 times 6. And then, 12 goes
into 12 of course. 1 times 12. So that's the factors of 12. Let's write the factors of 8. Well, 1 goes into 8. 2 goes into 8. 3 does not go into 8. 4 does go into 8. And then the last factor,
pairing up with the 1 is 8. So now we've written all
the factors of 12 and 8. So let's figure out what the
common factors of 12 and 8 are. Well, they both have the
common factor of 1. And that's really
not so special. Pretty much every whole
number or every integer has the common factor of 1. They both share the common
factor 2 and they both share the common factor 4. So we're not just interested in
finding a common factor, we're interested in finding the
greatest common factor. So all the common
factors are 1, 2 and 4. And what's the
greatest of them? Well, that's pretty easy. It's 4. So the greatest common
factor of 12 and 8 is 4. Let me write that down
just for emphasis. Greatest common factor
of 12 and 8 equals 4. And of course, we could have
just as easily had said, the greatest common divisor
of 12 and 8 equals 4. Sometimes it does
things a little funny. Let's do another problem. What is the greatest common
divisor of 25 and 20? Well, let's do it the same way. The factors of 25? Well, it's 1. 2 doesn't go into it. 3 doesn't go into it. 4 doesn't go into it. 5 does. It's actually 5 times 5. And then 25. It's interesting that
this only has 3 factors. I'll leave you to think about
why this number only has 3 factors and other numbers
tend to have an even number of factors. And then now we do
the factors of 20. Factors of 20 are 1,
2, 4, 5, 10, and 20. And if we just look at this by
inspection we see, well, they both share 1, but that's
nothing special. But they both have the
common factor of? You got it-- 5. So the greatest common divisor
or greatest common factor of 25 and 20- well, that equals 5. Let's do another problem. What is the greatest common
factor of 5 and 12? Well, factors of 5? Pretty easy. 1 and 5. That's because it's
a prime number. It has no factors other
than 1 and itself. Then the factors of 12? 12 has a lot of factors. It's 1, 2, 3, 4, 6, and 12. So it really looks like only
common factor they share is 1. So that was, I guess, in some
ways kind of disappointing. So the greatest common
factor of 5 and 12 is 1. And I'll throw out some
terminology here for you. When two numbers have a
greatest common factor of only 1, they're called
relatively prime. And that kind of makes sense
because a prime number is something that only has 1
and itself as a factor. And two relatively prime
numbers are numbers that only have 1 as their
greatest common factor. Hope I didn't confuse you. Let's do another problem. Let's do the greatest common
divisor of 6 and 12. I know 12's coming up a lot. I'll try to be more creative
when I think of my numbers. Well, the greatest common
divisor of 6 and 12? Well, it's the factors of 6. Are 1, 2, 3, and 6. Factors of 12: 1, 2, 3--
we should have these memorized by now. 3, 4, 6, and 12. Well, it turns out 1 is a
common factor of both. 2 is also a common
factor of both. 3 is a common factor of both. And 6 is a common
factor of both. And of course, what's the
greatest common factor? Well, it's 6. And that's interesting. So in this situation the
greatest common divisor-- and I apologize that I keep switching
between divisor and factor. The mathematics community
should settle on one of the two. The greatest common divisor
of 6 and 12 equals 6. So it actually equals
one of the numbers. And that makes a lot of
sense because 6 actually is divisible into 12. Well, that's it for now. Hopefully you're ready to do
the greatest common divisor or factor problems. I think I might make another
module in the near future that'll give you more
example problems.