# 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

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

https://gmatclub.com/forum/if-x-and-y-are-positive-integers-which-of-the-following-74924.html

### Great concept sir.

Great concept sir.
Would you explain option 20y please?

### D) If 20y is the greatest

D) If 20y is the greatest common DIVISOR of 35x and 20y, then it must be true that 35x/20y and 20y/20y are INTEGERS

20y/20y = 1, which is definitely an integer. Done!

What about 35x/20y? Can this ever be an integer?
You bet.
35x/20y = 7x/4y = (7/4)(x/y)
So, if x = 4 and y = 7, then 35x/20y = 140/140 = 1, which is an integer.
Also notice that, x = 4 and y = 7, then 35x = 140 and 20y = 140
The greatest common divisor of 140 and 140 is 140, and 140 = 20y (when y = 7)

So, 20y COULD be the greatest common divisor of 35x and 20y

Cheers,
Brent

Gotcha!

### x and y are positive integers

x and y are positive integers. If the greatest common divisor of 2x and 2y is 30, what is the greatest common divisor of x and 2y?

1) y is odd
2) x is odd
Didn't understand that " prime factorization of x contain a 2"?

Let's start at the point in the solution where we have:
x = (3)(5)(?)(?)(?)
y = (3)(5)(?)(?)(?)
Earlier in the solution we concluded that, "since we already know that there is no additional overlap beyond the ONE 3, and ONE 5, we can conclude that the greatest common divisor (GCD) of x and y is 15.

We then went on to examine the prime factorizations of x and 2y:
x = (3)(5)(?)(?)(?)
2y = (2)(3)(5)(?)(?)(?)

Notice that, if the prime factorization of x INCLUDES a 2, then we have...
x = (3)(5)(2)(?)(?)
2y = (2)(3)(5)(?)(?)(?)
... In which case the greatest common divisor of x and 2y = (2)(3)(5) = 30

Conversely, if the prime factorization of x does NOT include a 2, then we have...
x = (3)(5)(?)(?)(?)
2y = (2)(3)(5)(?)(?)(?)
...in which case the greatest common divisor of x and 2y = (3)(5) = 15

Does that help?

### hi Brent,

hi Brent,

My question maybe too basic but I would like to know the use of finding gcd of a number?

### There's no such thing as the

There's no such thing as the greatest common divisor (GCD) of ONE number.

Did you mean "prime factorization" of a number?
If so, this video covers that: https://www.gmatprepnow.com/module/gmat-integer-properties/video/825

If you're looking for how to calculate the greatest common divisor of TWO values, watch this video: https://www.gmatprepnow.com/module/gmat-integer-properties/video/833

### Hi Brent,

Hi Brent,

I’m sorry that was a typo. I have watched your videos on GCD and I am clear on that. However, I was actually asking about the areas of application for GCD. Like, what is the point in finding GCD of two numbers ?

### Q: What is the point in

Q: What is the point in finding the GCD of two numbers?

There aren't many ways to test greatest common divisor on the GMAT.
The linked questions above (in the Reinforcement Activities box) are a few examples of how you might be tested on this concept.

Cheers,
Brent