zaarathelab wrote:If w, x, y and z are integers such that w/x and y/z are integers, is w/x + y/z odd?
(1) wx + yz is odd
(2) wz + yx is odd
What is the quickest way to solve this?
Before we evaluate the two statements, we should examine how the question stem can be rephrased.
w/x + y/z = (wz + xy)/xz.
Since w/x and y/z are integers, their sum (w/x + y/z) is an integer.
Thus, (wz + xy)/xz must also be an integer.
The question becomes:
Is integer w/x + y/z -- which can be rephrased as (wz + xy)/xz -- odd?
Statement 1: wx + yz = odd.
Let w=1, x=1, y=2 and z=2, so that wx + yz = 1*1 + 2*2 = 5.
Is w/x + y/z odd?
NO, since 1/1 + 2/2 = 2.
Let w=1, x=1, y=6, and z=3, so that wx + yz = 1*1 + 6*3 = 19.
Is w/x + y/z odd?
YES, since 1/1 + 6/3 = 3.
INSUFFICIENT.
Statement 2: wz + xy = odd.
Please note the values highlighted in red:
Just as
10/2=5 is a factor of
10, and
12/3=4 is a factor of
12, so too is
(wz + xy)/xz a FACTOR of
wz + xy.
Since wz + xy is odd, all of its factors must be odd.
Thus, (wz + xy)/xz must be odd.
SUFFICIENT.
The correct answer is
B.
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