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### Course: 6th grade (Eureka Math/EngageNY) > Unit 2

Lesson 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

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# Greatest common factor examples

The greatest common factor (GCF) of a set of numbers is the largest factor that all the numbers share. For example, 12, 20, and 24 have two common factors: 2 and 4. The largest is 4, so we say that the GCF of 12, 20, and 24 is 4. GCF is often used to find common denominators. Created by Sal Khan.

## Want to join the conversation?

- Why does multiplying the common prime factors work to show the greatest common divisor? How come there is no greater number besides the answer?(60 votes)
- for example the question is gcd(8,24)

8 = 2*2*2

24 = 2*2*2*3

there are 3 two in both 8 and 24. so lets simplify all the common factors:

2*2*2 = 2*2*2

8=8

gcd = 8(24 votes)

- The GCF (or GCD) is very useful in simplifying complex math problems when you move on to more advanced material. Sometimes, it can make the difference between being able to solve the problem and not being able to solve them.

For this level of math, it helps you reduce fractions to their simplest form, thus helping you to better understand the numbers involved. For example, would you realize, without simplifying that 226/791 is the same thing as 2/7?

so if you are dealing with fractions... also this is the base of many other (used in life) advanced math problems.

so fractions and it is the basis of many other advanced math problems you will use more in life.(35 votes) - Yes, there is the difference between GCF and GCD is that GCF is the greatest common factor and GCD is the greatest common denominator. A factor is a number or quantity that when multiplied with another produces a given number or expression. A denominator is the number below the line in a common fraction; a divisor.

I hope this helps!:)(32 votes) - can't you just think of 7 as a prime number and apply the gcd as 1? is 1 the only factor other than the number itself for every prime number? does prime factorization work for every gcd?(9 votes)
- Prime definition is whole number which is not divisible by any other whole number except itself and 1, so yes to the second question. 1st question is yes as long as the other number is not a factor of 7. and yes prime factorization will always work to see what is common between two numbers.

It helps to learn divisibility rules

All even numbers (except 2) has 2 as a factor

If the last two digits (tens and ones) are divisible by 4, whole number is divisible by 4 (or divide by 2 twice)

If the last 3 digits (hundreds, tens and ones) are divisible by 8, whole number is divisible by 8 (or 2 three times).

If the sum of the digits add up to a factor of 3, it is divisible by 3, and same thing if add up to 9, divisible by 9 (or 3 twice)

6 is a combo of 2 and 3.

If the ones digit is 5 or 0, it is divisible by 5.

With any zeroes on end, you have equal number of 5s and 2s

This covers all numbers up to 10 except 7.(13 votes)

- Because a lot of people do not get it, I will expand apon the idea.

GCF is saying that what number can go into*__ and ___*?

For example, let’s take 24 and 36

So we write out the factors:

24: 1,2,12,24

36:1,3,12,36

Now we find what they both have in common? Try and figure it out yourself.

Did you get it? It’s 12. So 12 is our GCF or GCD(10 votes)- yes but you for got you have to have the greatest common factor you said just a common factor(6 votes)

- Around4:23why did Sal chose 3 over the 2? Two is the smallest number that goes into 21.(3 votes)
- well actually. since two is an even number, it cannot add up to an odd number so in this case three was actually the smallest number(10 votes)

- Is the greatest common divisor another name for the greatest common factor??(6 votes)
- Good question! This question confused me for a bit too. Anyway, Yes, it is.(3 votes)

- Are GCF (greatest common factor) and GCD (greatest common divisor) the same thing?(5 votes)
- Yes... they are and people can sometimes misinterpret them(4 votes)

- they should put the grade level next to the link so you know your on the right grade level.Because I'm in 4th grade and this is 6th.(6 votes)
- is there actually a prime number? because 7 is 3.5 X 2

1,3.5,7(4 votes)

## Video transcript

We're asked, what is the
greatest common divisor of 20 and 40? And they just say,
another way to say this is the GCD, or greatest
common divisor, of 20 of 40 is equal to question mark. And greatest common divisor
sounds like a very fancy term, but it's really
just saying, what is the largest number that is
divisible into both 20 and 40? Well, this seems like a pretty
straightforward situation, because 20 is actually
divisible into 40. Or another way to
say it is 40 can be divided by 20
without a remainder. So the largest
number that is a-- I guess you could say-- factor of
both 20 and 40 is actually 20. 20 is 20 times 1,
and 40 is 20 times 2. So in this situation,
we don't even have to break out our paper. We can just write 20. Let's do a couple more of these. So we're asked, what is
the greatest common divisor of 10 and 7? So let's now break out
our paper for this. So our greatest common
divisor of 10 and 7. So let me write that down. So we have 10. We want to think about what
is our GCD of 10 and 7? And there's two ways that
you can approach this. One way, you could literally
list all of the factors-- not prime factors, just
regular factors-- of each of these numbers and figure
out which one is greater or what is the largest
factor of both. So, for example, you could
say, well, I got a 10, and 10 can be expressed
1 times 10 or 2 times 5. 1, 2, 5, and 10. These are all factors of 10. These are all, we could
say, divisors of 10. And sometimes this is called
greatest common factor. Seven-- what are
all of its factors? Well, 7 is prime. It only has two
factors-- 1 and itself. So what is the
greatest common factor? Well, there's only one
common factor here, 1. 1 is the only common factor. So the greatest common
factor of 10 and 7, or the greatest common divisor,
is going to be equal to 1. So let's write that down. 1. Let's do one more. What is the greatest common
divisor of 21 and 30? And this is just another
way of saying that. So 21 and 30 are the two
numbers that we care about. So we want to figure out
the greatest common divisor, and I could have written
greatest common factor, of 21 and 30. So once again, there's
two ways of doing this. And so there's the way I did
the last time where I literally list all the factors. Let me do it that
way really fast. So if I say 21, what
are all the factors? Well, it's 1 and
21, and 3, and 7. I think I've got all of them. And 30 can be written as 1 and
30, 2 and 15, and 3-- actually, I'm going to run out of them. Let me write it this way so
I get a little more space. So 1 and 30. 2 and 15. 3 and 10. And 5 and 6. So here are all of
the factors of 30. And now what are
the common factors? Well, 1 is a common factor. 3 is also a common factor. But what is the
greatest common factor or the greatest common divisor? Well, it is going to be 3. So we could write 3 here. Now, I keep talking
about another technique. Let me show you the
other technique, and that involves the
prime factorization. So if you say the prime
factorization of 21-- well, let's see, it's divisible by 3. It is 3 times 7. And the prime factorization
of 30 is equal to 3 times 10, and 10 is 2 times 5. So what are the
most factors that we can take from both 21 and
30 to make the largest possible numbers? So when you look at the
prime factorization, the only thing that's common
right over here is a 3. And so we would say that
the greatest common factor or the greatest common
divisor of 21 and 30 is 3. If you saw nothing in
common right over here, then you say the greatest
common divisor is one. Let me give you another
interesting example, just so that we can get a
sense of things. So let's say these two
numbers were not 21 and 30, but let's say we care about
the greatest common divisor not of 21, but let's
say of 105 and 30. So if we did the prime
factorization method, it might become a
little clearer now. Actually figuring out, hey,
what are all the factors of 105 might be a little bit
of a pain, but if you do a prime factorization,
you'd say, well, let's see, 105-- it's divisible
by 5, definitely. So it's 5 times 21,
and 21 is 3 times 7. So the prime
factorization of 105 is equal to-- if I write them
in increasing order-- 3 times 5 times 7. The prime factorization of
30, we already figured out is 30 is equal to
2 times 3 times 5. So what's the most
number of factors or prime factors that
they have in common? Well, these two both have a
3, and they both have a 5. So the greatest common factor
or greatest common divisor is going to be a
product of these two. In this situation,
the GCD of 105 and 30 is 3 times 5, is equal to 15. So you could do it either way. You could just list out the
traditional divisors or factors and, say, figure
out which of those is common and is the greatest. Or you can break it down
into its core constituencies, its prime factors,
and then figure out what is the largest set
of common prime factors, and the product
of those is going to be your greatest
common factor. It's the largest number that
is divisible into both numbers.