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If \(a\) and \(b\) are positive integers, what is the greatest common divisor of \(a\) and \(b\)?

Notice that two statements together are obviously sufficient to answer the question. When you see such question you should be extremely cautious when choosing C for an answer. Chances are that the question is a "C trap" question ("C trap" is a problem which is VERY OBVIOUSLY sufficient if both statements are taken together).

(1) \(a + 3b = 61\).

Let the greatest common divisor of \(a\) and \(b\) be \(d\), then \(a=md\) and \(b=nd\), for some positive integers \(m\) and \(n\). So, we'll have \((md)+3(nd)=d(m+3n)=61\). Now, since 61 is a prime number (61=1*61) then \(d=1\) and \(m+3n=61\) (vice versa is not possible because \(m\) and \(n\) are positive integers and therefore \(m+3n\) cannot equal to 1). Hence we have that the GCD(x, y)=d=1. Sufficient.

(2) \(5a - b = 1\) --> \(5a=b+1\).

\(5a\) and \(b\) 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, \(5a\) and \(b\) don't share any common factor but 1, thus \(a\) and \(b\) also don't share any common factor but 1. Hence, the GCD(x, y) is 1. Sufficient.

Answer: D.

Try NEW divisibility PS question.
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Re: If a and b are positive integers, what is the greatest commo [#permalink]

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12 Jun 2014, 07:32

1

This post received KUDOS

D I am not sure if there is any standard approach for these...Just hit out some numbers

For I: I see that when A is odd say 3 then B is even 52..As B is an integer..the value of A will go down by 3 slots for each increase in B..So there is no case such that A and B will have any common factor other than 1 For II: You can again try a few..you can see that it is again 1...
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Appreciate the efforts...KUDOS for all Don't let an extra chromosome get you down..

If \(a\) and \(b\) are positive integers, what is the greatest common divisor of \(a\) and \(b\)?

Notice that two statements together are obviously sufficient to answer the question. When you see such question you should be extremely cautious when choosing C for an answer. Chances are that the question is a "C trap" question ("C trap" is a problem which is VERY OBVIOUSLY sufficient if both statements are taken together).

(1) \(a + 3b = 61\).

Let the greatest common divisor of \(a\) and \(b\) be \(d\), then \(a=md\) and \(b=nd\), for some positive integers \(m\) and \(n\). So, we'll have \((md)+3(nd)=d(m+3n)=61\). Now, since 61 is a prime number (61=1*61) then \(d=1\) and \(m+3n=61\) (vice versa is not possible because \(m\) and \(n\) are positive integers and therefore \(m+3n\) cannot equal to 1). Hence we have that the GCD(x, y)=d=1. Sufficient.

(2) \(5a - b = 1\) --> \(5a=b+1\).

\(5a\) and \(b\) 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, \(5a\) and \(b\) don't share any common factor but 1, thus \(a\) and \(b\) also don't share any common factor but 1. Hence, the GCD(x, y) is 1. Sufficient.

Answer: D.

Kudos points given to correct solutions above.

Try NEW divisibility PS question.
_________________

If \(a\) and \(b\) are positive integers, what is the greatest common divisor of \(a\) and \(b\)?

(1) \(a + 3b = 61\)

(2) \(5a - b = 1\)

Kudos for a correct solution.

Let k be the greatest common divisor of a and b such that a=kx and b=ky, where x and y are co-prime positive integers

St. 1; a+3b=61; kx+3ky=61; x+3y=(61/k) now since 61 is a prime number therefore maximum value of k is 1 hence sufficient

st.2 5a-b=1 5kx-ky=1 5x-y=(1/k) since x and y are integers, therefore 5x-y will only be integer if k=1

hence sufficient

therefore answer should be D

in st2 we can also say that bis multiple of 3. so this can be odd or even. so at a time b can be odd then a is even. or if b is even then a is odd. so GCD of a odd and even no. is 1. st2 is straight . numbers are co-prime

Re: If a and b are positive integers, what is the greatest commo [#permalink]

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15 Nov 2017, 21:31

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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