BTGmoderatorLU wrote:Source: Economist GMAT
If \(a\) and \(b\) are integers, and m is an even integer, is \(\frac{ab}{4}\) an integer?
1) \(a+b\) is even.
2) \(\frac{m}{ab}\) is an odd integer.
The OA is C
Let's take each statement one by one.
1) \(a+b\) is even.
Case 1: Say a = b = 1, thus, we have a + b = 2, an even number. But ab/4 = 1/4 is not an integer. The answer is no.
Case 2: Say a = b = 2, thus, we have a + b = 4, an even number. But ab/4 = 4/4 = 1 is an integer. The answer is yes.
No unique answer. Insufficient.
2) \(\frac{m}{ab}\) is an odd integer.
Since m is an even and m/ab is an odd integer, ab must be even integer.
Case 1: Say a = 2 and b = 1, thus, ab = 2*1 = 2, an even number. But ab/4 = 2/4 =1/2 is not an integer. The answer is no.
Case 2 Say a = 2 and b = 2, thus, ab = 2*2 = 4, an even number. But ab/4 = 4/4 =1 is an integer. The answer is yes.
No unique answer. Insufficient.
(1) and (2) together
So, we have (a + b) as well as ab as even integers. Thus, both a and b must be even. Say a = 2n and b = 2m, where n and m are integers. Thus. ab = 2n*2m = 4mn; thus. ab/4 = (2n*2m)/4 = mn, an integer. Sufficient.
The correct answer:
C
Hope this helps!
-Jay
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