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

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Math Expert
Joined: 02 Sep 2009
Posts: 58445
If a and b are positive integers, what is the greatest commo  [#permalink]

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12 Jun 2014, 07:37
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56% (02:08) correct 44% (01:57) wrong based on 681 sessions

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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.

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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, 07:37
9
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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.

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, 10:58
5
1
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

##### General Discussion
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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, 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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Joined: 02 Sep 2009
Posts: 58445
Re: If a and b are positive integers, what is the greatest commo  [#permalink]

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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.

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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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

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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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!
Math Expert
Joined: 02 Sep 2009
Posts: 58445
Re: If a and b are positive integers, what is the greatest commo  [#permalink]

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12 Nov 2016, 03:39
1
1
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!

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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20 Nov 2018, 22:21
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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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