6th grade (Eureka Math/EngageNY)
Course: 6th grade (Eureka Math/EngageNY) > Unit 2Lesson 4: Topic D: Number theory—thinking logically about multiplicative arithmetic
- Divisibility tests for 2, 3, 4, 5, 6, 9, 10
- Recognizing divisibility
- The why of the 3 divisibility rule
- The why of the 9 divisibility rule
- Divisibility tests
- Intro to even and odd numbers
- Greatest common factor examples
- Greatest common factor explained
- Greatest common factor
- Greatest common factor review
- Least common multiple
- Least common multiple: repeating factors
- Least common multiple of three numbers
- Least common multiple
- Least common multiple review
- GCF & LCM word problems
- GCF & LCM word problems
Greatest common factor explained
Here's a nice explanation of greatest common factor (or greatest common divisor) along with a few practice example exercises. Let's roll. Created by Sal Khan.
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- What about bigger numbers like 118 and 204(90 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 )
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.(162 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.(41 votes)
- while I understand the problems that Sal uses in this video, I do not understand their relation to the problems in the attached section. The first question I was faced with was "#'s divisible by both 15 and 8 are also divisible by which of the following: 21,55,35,60,33". I'm sure with the right explanation, this is very simple, but this specific video stops short. The GCF or GCD of (15,8) is 1. I understand that, but that's not what was asked. Is this question in the wrong section?(12 votes)
- Ok Sal does not explain is explicitly in the video, but it is related. ok so first you have to understand what the problem is asking. The problem is asking for a number(the choices) that have the same factor. So 15's factors are 1 and 5, 15. And then for 8 the factors are 1, 2, 4, 8. So which of the choices is divisible by 1, 2, 4, , 1, 5, 8, 15.
21: is not divisible by 2,5,4 8...
While 60 is divisible by all of the factors(1, 2, 4, 5, 8, 15)
The key is to understand the questions!
Hope that helps
- I need to know how to do it with bigger number that's what I do in the exercise.(15 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
so the factors of 318 are
41 -- prime
so the factors are
Line up the factors
so the greatest common divisor of 492 and 318 will be
2 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(21 votes)
- ls there any numer that has the factors 1 2 3 4 5 6 7 8 and 9(11 votes)
- There can be more, of course, if you multiply 2/3/4/5/6/7/8/9 to 362880.(7 votes)
- Is GCM a concept in math? I don't know if my teacher said that accidentally instead of GCF.(10 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).
- I don't get it. What is the difference between GCD and GCF?(7 votes)
- They are the same(5 votes)
- I was trying to complete the 'Divisibility' Exercise. I was unable to get the correct answers. This was the video it had me watch to help me, yet it does not apply to the 'Divisibility' Exercises. What should I watch for help with the 'Divisibility' Exercises?(6 votes)
- I believe the lowest common multiple of 1, 2, 3, 4, 5, 6, 7, 8 & 9 is 15,120.
Think of it like this: 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 = 362,880
But if a number has 9 or 6 as a factor, it automatically has 3 as a factor as well, because 3 is a factor of 9 and 6; so we can remove 3 from that list. And if it has 8 as a factor, it automatically has 4 and 2 as factors as well, so we can remove 4 and 2 from that list.(3 votes)
- I'm not really shore what is the difference of (GCD) and (GCF)?(8 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.(5 votes)
- Hi Sal and the entire Khan Academy!! :)
Can you please explain divisibility. :) I'm doing the practice test on divisibility and I've watched the GCD video (which I understand), but it doesn't explain this question: "All numbers divisible both 12 and 6 are also divisible by which of the following? 33, 22, 14, 4, and 21. I figured 3 is divisible by both 12 and 6 and 21 and picked 21, but I was wrong.(8 votes)
- Just in case my link is deleted from the comments again:
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.