# Lesson: Greatest Common Divisor (GCD)

## Comment on Greatest Common Divisor (GCD)

### Hi Brent,

Hi Brent,

I'm probably missing something, but I think I've found a situation where the tactic explained in this video leads to the incorrect answer. Consider the question, "What is the greatest common divisor of 300 and 240?" The common prime factors of these two numbers are 5, 3, and 2, which, when multiplied together equal 30. However, the greatest common divisor of 300 and 240 is 60. Could you possibly help me understand where I'm making a mistake?

Ben ### We must consider numbers

We must consider numbers sharing a prime factor more than once. 300 and 240 have two 2's in common.

300 = (2)(2)(3)(5)(5)
240 = (2)(2)(2)(2)(3)(5)
So, 300 and 240 have two 2's, one 3, and one 5 in common.
So, the GCF = (2)(2)(3)(5) = 60

### That makes sense. Thank you

That makes sense. Thank you for clarifying!

-Ben

### Hi Brent,

Hi Brent,

Below link has one of above question

https://gmatclub.com/forum/x-and-y-are-positive-integers-if-the-greatest-common-divisor-of-3x-237380.html

Below is snippet from your solution,

So,
2x = (2)(2)(?)(?)(?)(?)
2y = (2)(2)(?)(?)(?)(?)
This tells us that the greatest common divisor (GCD) of 2x and 2y =(2)(2) = 4.

-How can we assume that there is no common factor in values represented by ???

i.e.

2x=2*2*a
2y=2*2*b

a and b can have common factor isn't it?

Though I have got GCD concept well but I am struggling to solve questions, your advice will help. Let's step back for a second.
If I say that the GCD of x and y is 2, that means the ONLY thing that the prime factorizations of x and y have in common is ONE 2.

In other words, if we examine the prime factorizations of x and y, we get:
x = (2)(?)(?)(?)(?)...
y = (2)(?)(?)(?)(?)...
IMPORTANT: the ?'s represent other possible primes in the factorizations. HOWEVER, there is no additional overlap beyond the ONE 2.

For example, consider these prime factorizations:
x = (2)(3)(3)(5)(7)(7)
y = (2)(2)(2)(19)
Here, x and y have ONE 2 in common. So, the GCD of x and y is 2

Another example:
x = (2)(2)(11)(13)
y = (2)(5)(5)(5)
Here, x and y have ONE 2 in common. So, the GCD of x and y is 2

So later, when we multiply x and y by 2, we get:
2x = (2)(2)(?)(?)(?)(?)
2y = (2)(2)(?)(?)(?)(?)
Once again, the ?'s represent other possible primes in the factorizations.
HOWEVER, since there is no additional overlap beyond the TWO shared 2's, the GCD of 2x and 2y must be 4

Does that help?

### yes makes sense. Thanks a lot

yes makes sense. Thanks a lot.

### https://gmatclub.com/forum/x

https://gmatclub.com/forum/x-and-y-are-positive-integers-if-the-greatest-common-divisor-of-3x-237380.html

I don't know why I'm having trouble understanding your approach to the problem - it seems to make perfect sense. The way I approach GCD and LCM questions is to put it into a table and factor out the primes. For instance,

18 2^1 3^2

24 2^3 3^1

Look at the smallest common exponents and that gets 2^1 times 3^1 = 6

How do I reconcile this method with yours? You have to take a concrete example (as you have done for 18 and 24) and then apply the same principles to the more abstract variables 3x and 3y.

This is what I have done in my solution at https://gmatclub.com/forum/x-and-y-are-positive-integers-if-the-greatest...

Take a look at that solution, and let me know if you need any clarification.

Cheers,
Brent

### Hi! Brent, I could only solve

Hi! Brent, I could only solve last 2 questions given above which were more difficult than the ones before these:
1.) https://gmatclub.com/forum/x-and-y-are-positive-integers-if-the-greatest-common-divisor-of-2x-237473.html
2.) https://gmatclub.com/forum/x-and-y-are-positive-integers-if-the-greatest-common-divisor-of-3x-237380.html

Am I missing something in easier questions, whereas I'm able to correctly solve difficult ones in time. ### For the first 3 linked

For the first 3 linked questions, it's very useful to be able to identify values that satisfy the given information. This is a useful skill for solving many Integer Properties questions. So, you might want to work on that.

For more on this, see: https://www.gmatprepnow.com/module/gmat-integer-properties/video/845

Cheers,
Brent

### Hi Brent, in the video you

Hi Brent, in the video you mention about listing the greatest common factor.

Is there technic on how to list out the common factors,
other than from starting and ending > (1, ...... ,56).

Hope to hear from you, thank you!
Wuan ### Great question!

Great question!

Unfortunately, there isn't a technique faster than listing them in pairs (that multiply to the given value).
For example, for 56 we get: 1 & 56, 2 & 28, 4 & 14 etc

That said, it may help (a little) to first find the prime factorization of the number. This can help you list the values in pairs.

For example, 56 = 2 x 2 x 2 x 7
So, we can see that 3 is not a factor, 5 is not a factor, etc.
We can also easily see that 2 is a factor, 4 is a factor, etc.

Cheers,
Brent