If x and y are positive integers, what is the greatest common divisor

Question : GCD of x and y = ?

Statement 1: 2x + y = 73

This statement can give us multiple solutions of x and y but the important part is to notice the value of GCD in each case e.g.
(y=1, x=36) GCD = 1
(y=3, x=35) GCD = 1
(y=5, x=34) GCD = 1
(y=7, x=33) GCD = 1
(y=9, x=32) GCD = 1... and so on...

Finally we realize that instead of multiple solutions of x and y, their GCD is consistently 1,
Hence SUFFICIENT

Statement 2: 5x – 3y = 1

(y=3, x=2) GCD = 1
(y=8, x=5) GCD = 1
(y=13, x=8) GCD = 1
(y=18, x=11) GCD = 1
(y=23, x=14) GCD = 1... and so on...

Finally we realize that instead of multiple solutions of x and y, their GCD is consistently 1,
Hence SUFFICIENT

Answer: Option

Point to Learn: In all such equations with two variable you can realize that the solutions have a harmony i.e. value of variable x changes by co-efficient of y and value of y changes by co-efficient of x and this relation holds true in all such equation where the GCD of co-efficients of x and y is 1.

If there is some common factor among co-efficients of x and y then cancel the common factor and the rule holds true in those cases with modified equation.

Here is what i did => i made the pairs of values and saw the pattern and then compiled that D is correct
still not able to see a proper solution on this page
some are quoting algebra and some are doing by values putting
maybe chetan2u will be helpful here..
Any other methods?

Have you checked out Bunuel's solution on the first page?
if-x-and-y-are-positive-integers-what-is-the-greatest-128552.html#p1053416

It explains the best way to deal with this question.

If you want to avoid algebra, think about it like this:

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

1) 2x + y = 73
Say, x and y have a common factor f other than 1. If that is the case, you should be able to take f common out of the two terms on left hand side. So you will get
f*something = 73
But 73 cannot be written as product of two numbers other than 1 and itself. So f MUST BE 1. Hence greatest common divisor of x and y MUST BE 1. Sufficient

2) 5x – 3y = 1
Here, 5x and 3y are consecutive integers (since difference between them is 1). Consecutive integers can share no common factor other than 1. So 5x and 3y have no common factors. This means that x and y can have no common factors (other than 1) too. Else that factor would have been common between 5x and 3y too. Hence greatest common divisor of x and y MUST BE 1. Sufficient

Answer (D)

just a general thing .y is odd and x is even take y =15 and x=30 .GCD N.E. 1

so there's gotta be other approach just by saying y is odd and x even won't get GCD=1 always

i guess if this type of question encounters u better skip taking a hard guess .Dont waste time(BTW gmat won't give this type of problem involving so much calculations.the paper always play with tricks which you have to find out)

Bunuel one silly question but please help with this :

Through the above logic for point 2 ,how can we be sure that d =1 .What about d= 1/(5M-3N) and that being another value apart from 1

Thanks a ton for all the help!!!

We have \(d(5m-3n)=1\). Both d and (5m-3n) are positive integers, there is only one value of d possible, d = 1.

Dear GMATinsight

Can you please elaborate the red bold statements with examples? I do not grasp the coefficient issue here.

Thanks in advance

Solution:

Statement 1:Sufficient
Test cases:
x = 2, y = 69, GCD = 1
x = 3, y = 67, GCD = 1
x = 4, y = 65, GCD = 1

Statement 2:Sufficient
Test cases:
x = 2, y = 3, GCD = 1
x = 3, y = 4, GCD = 1
x = 5, y = 8, GCD = 1

The answer is D.

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