Lesson: Greatest Common Divisor (GCD)

Comment on Greatest Common Divisor (GCD)

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?

Many thanks in advance,
Ben
gmat-admin's picture

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 for clarifying!

-Ben

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.
gmat-admin's picture

Question link: https://gmatclub.com/forum/x-and-y-are-positive-integers-if-the-greatest...

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.

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