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What is the greatest common factor of positive integers a

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

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New post Updated on: 16 Dec 2012, 08:12
3
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A
B
C
D
E

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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, 15:30.
Last edited by Bunuel on 16 Dec 2012, 08:12, edited 2 times in total.
Added the OA
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Re: GCF of integers a & b  [#permalink]

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New post 27 Jan 2012, 15:50
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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.

Answer: C.

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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New post 27 Jan 2012, 15:53
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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New post 27 Jan 2012, 15:56
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Re: GCF of integers a & b  [#permalink]

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New post 25 May 2012, 19:59
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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New post 15 May 2014, 03:42
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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New post 15 May 2014, 05:44
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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]

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New post 11 Jul 2016, 08:38
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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Re: What is the greatest common factor of positive integers a  [#permalink]

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New post 11 Jul 2016, 08:39
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

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

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New post 29 Aug 2016, 09:33
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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New post 14 Oct 2016, 23:09
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


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

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New post 27 Dec 2016, 05:57
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(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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New post 27 Sep 2018, 04:33
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 &nbs [#permalink] 27 Sep 2018, 04:33
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