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If a and b are positive integers, what is the greatest commo

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

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New post 12 Jun 2014, 07:37
3
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A
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C
D
E

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Question Stats:

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

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New post 12 Jun 2014, 07:37
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SOLUTION

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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New post 12 Jun 2014, 10:58
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Bunuel wrote:


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

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New post 12 Jun 2014, 08:32
1
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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Re: If a and b are positive integers, what is the greatest commo  [#permalink]

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New post 15 Jun 2014, 04:57
2
1
SOLUTION

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

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New post 14 Aug 2014, 01:53
manpreetsingh86 wrote:
Bunuel wrote:


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

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New post 11 Nov 2016, 20:18
Is there a compilation of such 700 level questions?

I really like these. Sometimes some of them are tricky.

Could someone please point me to any collection that you are aware of.

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

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New post 12 Nov 2016, 03:39
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manhasnoname wrote:
Is there a compilation of such 700 level questions?

I really like these. Sometimes some of them are tricky.

Could someone please point me to any collection that you are aware of.

Many thanks!


Please browse the site:
viewforumtags.php
150-hardest-and-easiest-questions-for-ds-204132.html
ds-question-directory-by-topic-and-difficulty-128728.html
150-hardest-and-easiest-questions-for-ps-204134.html
gmat-ps-question-directory-by-topic-difficulty-127957.html
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Re: If a and b are positive integers, what is the greatest commo  [#permalink]

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Re: If a and b are positive integers, what is the greatest commo   [#permalink] 20 Nov 2018, 22:21
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