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

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Joined: 02 Mar 2012
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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
1
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)
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Re: If x and y are positive integers, what is the greatest  [#permalink]

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02 Apr 2017, 02:30
Bunuel wrote:
carcass wrote:
Thanks bunuel.

From your explanation can we suppose to apply the reasoning from statement two to the first one or is not possible ?? would be a big mistake because 1 is NOT prime??'

I suppose you mean whether we can apply the reasoning from (1) to statement (2). Yes, we can:

(2) $$5x-3y=1$$ --> 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 $$5(md)-3(nd)=d(5m-3n)=1$$ --> $$d$$ is a factor of 1, so $$d$$ must equal 1. Sufficient.

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

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02 Apr 2017, 05:09
KARISHMA315 wrote:
Bunuel wrote:
carcass wrote:
Thanks bunuel.

From your explanation can we suppose to apply the reasoning from statement two to the first one or is not possible ?? would be a big mistake because 1 is NOT prime??'

I suppose you mean whether we can apply the reasoning from (1) to statement (2). Yes, we can:

(2) $$5x-3y=1$$ --> 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 $$5(md)-3(nd)=d(5m-3n)=1$$ --> $$d$$ is a factor of 1, so $$d$$ must equal 1. Sufficient.

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

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02 Apr 2017, 05:11
KARISHMA315 wrote:

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!!!

KARISHMA315,

Because factors are always integer. so D has to be taken as an integer only.

d= 1/(5M-3N) -- here (5M-3N) is nothing but 1. if (5M-3N) is anything other than 1 then D will not be an integer. and that would violate the rule "factors are always integer".
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Re: If x and y are positive integers, what is the greatest  [#permalink]

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18 Jun 2017, 22:28
Bunuel wrote:
zest4mba 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

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.

Hope it's clear.

Hello Buñuel,

I was wondering why can't we apply the same logic given by you for statement 2 in statement 1, i.e. even 2 and 3 don't have any common factor other than 1?

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

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17 Oct 2017, 07:30
AlexGenkins1234 wrote:
carcass 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

Here I'm not sure that the answer is C because is true that we need of both statement to find possible values for X and Y. Infact statement 1 and 2 we do not have values for the variables (can be everything).

But it seems to be a trap answer......

A very simple way to solve this problem will be:

1. 2x is even -> even +y=odd -> this mean y is odd -> even and odd GCD is 1 -> sufficient.
2. 5x-3y=1 -> 2 options:
- 5x is even -> [odd -3y=odd] and [x is even] -> 3y must be odd -> y must be odd -> X even & Y odd -> even and odd GCD is 1
- 5x is odd -> [odd-3y=odd] and [x is odd] -> 3y must be even -> Y must be even -> X is odd & Y is even -> even and odd GCD is 1

hence is both cases the GCD is 1. Sufficient.

This is not true: "even and odd GCD is 1"

Just take 6 (even) and 9 (odd). GDC is 3

Many other examples:
12, 15
14, 21
20, 25
etc.
Re: If x and y are positive integers, what is the greatest &nbs [#permalink] 17 Oct 2017, 07:30

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