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What is the greatest common factor of positive integers a  [#permalink]

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Question Stats: 71% (01:26) correct 29% (01:19) wrong based on 539 sessions

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What is the greatest common factor of positive integers a and b?

(1) a = b + 4
(2) b/4 is an integer

As OA is not given, I got C as an answer. Please let me know if its correct.

Statement 1 --> Clearly insufficient as we a & b can have several values.

Statement 2 --> b is a multiple of 4. i.e. 4, 8 , 12..... but as we don't know what a is, this statement is insufficient.

Combing the 2 -> Statement 1 will give values of a as 8, 12, 16 and from statement 2 we will have several values of b.

So considering a as 12 and b as 16

12 => 2^2 * 3
16 ==> 2^4

GCF = 2^2 = 4

Another value of a as 8 and b as 12

a = 2^3
b = 2^2 * 3

GCF = 2^2 = 4.

Therefore, both statement together (c) are sufficient to answer this question.

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Originally posted by enigma123 on 27 Jan 2012, 16:30.
Last edited by Bunuel on 16 Dec 2012, 09:12, edited 2 times in total.
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Re: GCF of integers a & b  [#permalink]

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What is the greatest common factor of positive integers a and b?

(1) a = b + 4. Clearly insufficient.
(2) b/4 is an integer --> $$b=4k$$ --> $$b$$ is a multiple of 4, though still insufficient as no info about a.

(1)+(2) $$a=b+4=4k+4=4(k+1)$$ --> useful property:
if $$a$$ and $$b$$ are multiples of $$k$$ and are $$k$$ units apart from each other then $$k$$ is greatest common divisor of $$a$$ and $$b$$. For example if $$a$$ and $$b$$ are multiples of 7 and $$a=b+7$$ then 7 is GCD of $$a$$ and $$b$$.

So, as we have that both $$a$$ and $$b$$ are multiples of 4 and are 4 units apart each other ($$a=b+4$$ ), then 4 is GCD of $$a$$ and $$b$$.

Or another way if you are not familiar with above property: we have $$a=4(k+1)$$ and $$b=4k$$, now $$k$$ and $$k+1$$ are consecutive integers thus they do not share any common factor but 1, which means that GCD of $$a$$ and $$b$$ is 4.

Similar questions (with explanation of this property):
gcd-of-a-b-126427.html
what-is-the-greatest-common-factor-of-x-and-y-1-x-and-y-are-109273.html
if-x-and-y-are-positive-integers-such-that-x-8y-12-what-101196.html

Hope it helps.
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Re: GCF of integers a & b  [#permalink]

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Thanks for the property. You told me about this before too, and I will make sure I remember it. Thanks again B _________________
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Re: GCF of integers a & b  [#permalink]

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enigma123 wrote:
Thanks for the property. You told me about this before too, and I will make sure I remember it. Thanks again B There is also another approach in above solution if one is not familiar with this property.
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Re: GCF of integers a & b  [#permalink]

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Bunuel wrote:

So, as we have that both $$a$$ and $$b$$ are multiples of 4 and are 4 units apart each other ($$a=b+4$$ ), then 4 is GCD of $$a$$ and $$b$$.

[b]Or another way if you are not familiar with above property:
we have $$a=4(k+1)$$ and $$b=4k$$, now $$k$$ and $$k+1$$ are consecutive integers thus they do not share any common factor but 1, which means that GCD of $$a$$ and $$b$$ is 4.

Hope it helps.

I want to quickly add to Bunuel's point, that this is the case because if gcd(x,y)=z then gcd(cx,cy)=cz, where c is some constant. so gcd(k,k+1)=1 (since they are consecutive intergers. Muliply by 4, gcd(4k,4(k+1))=4*1
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Re: What is the greatest common factor of positive integers a  [#permalink]

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what if i take a=1 and b = 4 , justifies both the equations 1 and 2 hence the answer in that case would be E.
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Re: What is the greatest common factor of positive integers a  [#permalink]

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SaudKhan wrote:
what if i take a=1 and b = 4 , justifies both the equations 1 and 2 hence the answer in that case would be E.

a = 1 and b = 4 does not satisfy the firs statement (a = b + 4 ).
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GRE 1: Q169 V154 Re: What is the greatest common factor of positive integers a  [#permalink]

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Nice Question..
Here combination statement works as it will make A and B as consecutive multiples of 4 =>GCD =4
C is sufficient
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GMAT 1: 750 Q49 V43 Re: What is the greatest common factor of positive integers a  [#permalink]

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What is the greatest common factor of positive integers a and b?
(1) a = b + 4
Insufficient
a=b+4
a-b=4
(8-4=4 GCF=1) OR (15-11=4;GCF = 4)

(2) b/4 is an integer
Insufficient
If b=12 then a=3+4=7 GCF=1
if b=16 then a=4+4=8 GCF=4

Merging Both
SUFFICIENT :- a and b become two consecutive multiple of 4 and will always have 4 as the GCF
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GMAT 1: 750 Q49 V43 Re: What is the greatest common factor of positive integers a  [#permalink]

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1
What is the greatest common factor of positive integers a and b?
(1) a = b + 4
Insufficient
a=b+4
a-b=4
(8-4=4 GCF=1) OR (15-11=4;GCF = 4)

(2) b/4 is an integer
Insufficient
If b=12 then a=3+4=7 GCF=1
if b=16 then a=4+4=8 GCF=4

Merging Both
SUFFICIENT :- a and b become two consecutive multiple of 4 and will always have 4 as the GCF

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FINAL GOODBYE :- 17th SEPTEMBER 2016. .. 16 March 2017 - I am back but for all purposes please consider me semi-retired.
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Re: What is the greatest common factor of positive integers a  [#permalink]

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enigma123 wrote:
What is the greatest common factor of positive integers a and b?

(1) a = b + 4
(2) b/4 is an integer

Please find the solution as attached
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Re: What is the greatest common factor of positive integers a  [#permalink]

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enigma123 wrote:
What is the greatest common factor of positive integers a and b?

(1) a = b + 4
(2) b/4 is an integer

Question : greatest common factor of positive integers a and b?

Statement 1: a = b + 4

@b=1, a=5 and GCD = 1
@b=2, a=6 and GCD = 2
NOT SUFFICIENT

Statement 2: b/4 is an integer
i.e b must be a multiple of 4 but
There is no information of a so
NOT SUFFICIENT

Combining the two statements
@b=4, a=8 and GCD = 4
@b=8, a=12 and GCD = 4
@b=12, a=16 and GCD = 4
@b=16, a=20 and GCD = 4
Since now a and b are two consecutive multiples of 4 therefore GCD will always be 4
SUFFICIENT

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Re: What is the greatest common factor of positive integers a  [#permalink]

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1
(1) INSUFFICIENT: a and b could be 12 and 8, with a greatest common factor of 4; or they could be 11 and 7, with a greatest common factor of 1.

(2) INSUFFICIENT: This statement tells us that b is a multiple of 4 but we have no information about a.

(1) AND (2) SUFFICIENT: Together, we know that b is a multiple of 4 and that a is the next consecutive multiple of 4. For any two positive consecutive multiples of an integer n, n is the greatest common factor of those multiples, so the greatest common multiple of a and b is 4. The correct answer is C.
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Re: What is the greatest common factor of positive integers a  [#permalink]

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But when it says b/4 is an integer, can b not be zero - wherein b/4 will yield 0, which is an integer?

And hence, the answer could be (E)?

Thanks
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Re: What is the greatest common factor of positive integers a  [#permalink]

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_________________ Re: What is the greatest common factor of positive integers a   [#permalink] 03 Oct 2019, 12:19
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