I believe that the problem intends to ask the following:
vipulgoyal wrote:How many DISTINCT PAIRS OF INTEGERS satisfy the equation (1/x) + (1/y) = 1/12?
a) 12
b) 6
c) 10
d) 16
e) 32
This problem seems beyond the scope of the GMAT.
That said, here's one approach:
1/x + 1/y = 1/12
(x+y)/xy = 1/12
12x + 12y = xy
12x - xy + 12y = 0
(12x - xy + 12y) + (x-12)(y-12) = (x-12)(y-12)
(12x - xy + 12y) + (xy - 12x - 12y + 144) = (x-12)(y-12)
144 = (x-12)(y-12).
Implication:
(x-12) and (y-12) are FACTOR PAIRS OF 144.
Options:
x-12 = 1 and y-12 = 144 --> x=13, y=156
x-12 = 2 and y-12 = 72 --> x=14, y=84
x-12 = 3 and y-12 = 48 --> x=15, y=60
x-12 = 4 and y-12 = 36 --> x=16, y=48
x-12 = 6 and y-12 = 24 --> x=18, y=36
x-12 = 8 and y-12 = 18 --> x=20, y=30
x-12 = 9 and y-12 = 16 --> x=21, y=28
x-12 = 12 and y-12 = 12 --> x=24, y=24.
Total options = 8.
If (x-12) and (y-12) are NEGATIVE FACTOR PAIRS of 144, we get another 8 options:
x-12 = -1 and y-12 = -144 --> x=11, y=-132
x-12 = -2 and y-12 = -72 --> x=10, y=-60
x-12 = -3 and y-12 = -48 --> x=9, y=-36
x-12 = -4 and y-12 =-36 --> x=8, y=-24
x-12 = -6 and y-12 = -24 --> x=6, y=-12
x-12 = -8 and y-12 = -18 --> x=4, y=-6
x-12 = -9 and y-12 = -16 --> x=3, y=-4.
Total options = 7.
Thus:
Total distinct integer pairs = 8+7 = 15.
Since 15 is not among the answer choices, I would ignore this problem.
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