What is the greatest common factor of positive integers a

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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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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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

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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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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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

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

Please find the solution as attached

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

(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.

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

Bunuel I understand how we get b is a multiple of 4 from statement 2. However how do you know a is a multiple of 4? Statement 1?

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b is a multiple of 4 and a = b + 4 = (a multiple of 4) + (a multiple of 4) = (a multiple of 4).
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What if b=0, as 0 is considered an integer as well, then the answer will be E, please let me know your thoughts

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

0 is not a positive integer.
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