gmatpapa
If c and k are distinct positive integers, is c divisible by k?
(1) \(2k > c\)
(2) \((k^2 + k)= c\)
Note that an integer \(a\) is a multiple of an integer \(b\) (integer \(a\) is a divisible by an integer \(b\)) means that \(\frac{a}{b}=integer\).
Given: \(c\) and \(k\) are
distinct positive integers. Question: is \(\frac{c}{k}=integer\)? Now, \(c\) and \(k\) are
distinct so if \(c\) is indeed divisible by \(k\) then \(\frac{c}{k}\) can ONLY be 2, 3, 4, ... basically any positive integer except 1 (again as \(c\) and \(k\) are distinct and positive).
(1) \(2k > c\) --> divide both sides of the inequality by \(k\) (we can safely do this as given that \(k\) is positive) --> \(2>\frac{c}{k}\) --> \(c\) is not divisible by \(k\). Sufficient.
(2) \((k^2 + k)= c\) --> \(k(k+1)=c\) --> \(\frac{c}{k}=k+1=integer\) --> \(c\) is divisible by \(k\). Sufficient.
Now, technically answer should be D, as
EACH statement ALONE is sufficient to answer the question.
But even though formal answer to the question is D (EACH statement ALONE is sufficient), this is not a realistic GMAT question, as:
on the GMAT, two data sufficiency statements always provide TRUE information and these statements never contradict each other.
So we can not have answer NO from statement (1) and answer YES from statement (2), as in this case statements would contradict each other.
Hope it's clear.