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Re: If x and y are positive integers, what is the greatest [#permalink]

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23 Aug 2010, 11:46

I am sure there is a clever answer to this, C obviously works... I note that from 2. X=(3y+1)/5, so y has to end in a 3 ie y= 3 13 23 etc. While x= 2 8 14 ie even, x even, can we say then that GCD = 1?

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

1. 2x + y = 73 2. 5x – 3y = 1

Say x and y were both divisible by some number d. Then 2x + y would certainly be a multiple of d (if you add two multiples of d, you always get a multiple of d). Now we know from statement 1 that 2x + y is the number 73, so if 2x+y is divisible by d, then 73 must be divisible by d. But 73 is prime, so d could only be 1 or 73. Clearly d can't be 73, since then 2x +y would not equal 73, so the only possible value of d is 1, and thus 1 is the only common divisor of x and y.

You can use the same logic for statement 2: If x and y are both multiples of d, then 5x - 3y would need to be a multiple of d. But 5x-3y = 1, so 1 is a multiple of d, and d must be 1.

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

1. 2x + y = 73 2. 5x – 3y = 1

This is a classic "C trap" question: "C trap" is a problem which is VERY OBVIOUSLY sufficient if both statements are taken together. When you see such question you should be extremely cautious when choosing C for an answer.

(1) \(2x+y=73\). Suppose GCD(x, y) is some integer \(d\), then \(x=md\) and \(y=nd\), for some positive integers \(m\) and \(n\). So, we'll have \(2(md)+(nd)=d(2m+n)=73\). Now, since 73 is a prime number (73=1*73) then \(d=1\) and \(2m+n=73\) (vice versa is not possible since \(m\) and \(n\) are positve integers and therefore \(2m+n\) cannot equal to 1). Hence we have that GCD(x, y)=d=1. Sufficient.

(2) \(5x-3y=1\) --> \(5x=3y+1\). So \(5x\) and \(3y\) are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1. For example 20 and 21 are consecutive integers, thus only common factor they share is 1. So \(5x\) and \(3y\) don't share any common factor but 1, thus \(x\) and \(y\) also don't share any common factor but 1. Hence, GCD(x, y) is 1. Sufficient.

Re: If x and y are positive integers, what is the greatest [#permalink]

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23 Aug 2010, 16:01

Bunuel, Is there a complete discussion on GCDs and LCMs on the forum? Can you please point me to the same? I am trying to recollect why is x y = GCD(x,y) x LCM(x,y)? Thanks
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Re: If x and y are positive integers, what is the greatest [#permalink]

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29 Aug 2014, 13:13

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Re: If x and y are positive integers, what is the greatest [#permalink]

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02 Jun 2015, 04:48

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

1. 2x + y = 73 2. 5x – 3y = 1

Another way of solving it .Both x and y are integers . From 1: x =(73-y)/2 .Since x is a integer it implies 73-y =even number .73 is Odd so y is also Odd .X is even so GCM will be 1.Sufficient

From 2 :5x-3y =1 .They are consecutive numbers i.e .odd-even or even -odd .so the GCM in this case =1 .Sufficient

Option D is correct .

Press Kudos if you like the solution.
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If x and y are positive integers, what is the greatest common divisor of x and y?

(1) 2x + y = 73 (2) 5x – 3y = 1

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

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.

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Re: If x and y are positive integers, what is the greatest [#permalink]

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20 Jun 2016, 01:21

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: If x and y are positive integers, what is the greatest [#permalink]

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20 Jun 2016, 05:12

manishkhare wrote:

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

1. 2x + y = 73 2. 5x – 3y = 1

Another way of solving it .Both x and y are integers . From 1: x =(73-y)/2 .Since x is a integer it implies 73-y =even number .73 is Odd so y is also Odd .X is even so GCM will be 1.Sufficient

From 2 :5x-3y =1 .They are consecutive numbers i.e .odd-even or even -odd .so the GCM in this case =1 .Sufficient

Option D is correct .

Press Kudos if you like the solution.

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)

gmatclubot

Re: If x and y are positive integers, what is the greatest
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20 Jun 2016, 05:12

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