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

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.

Answer: D.

Try NEW divisibility PS question.