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

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If x and y are positive integers, what is the greatest [#permalink]  23 Aug 2010, 10:27
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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
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Re: If x and y are positive integers, what is the greatest [#permalink]  23 Aug 2010, 10:46
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?

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Re: If x and y are positive integers, what is the greatest [#permalink]  23 Aug 2010, 11:58
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

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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Re: If x and y are positive integers, what is the greatest [#permalink]  23 Aug 2010, 12:00
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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 $$2m+n$$ can not 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.
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Re: If x and y are positive integers, what is the greatest [#permalink]  23 Aug 2010, 15:01
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]  29 Aug 2014, 12:13
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Re: If x and y are positive integers, what is the greatest   [#permalink] 29 Aug 2014, 12:13
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