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):
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Hope it helps.