If x, y, and z are integers and xy+z is an odd integer, is x an even integer?
1) xy+xz is an even integer
2) y+xz is an odd integer
Statement 1: xy + xz is even
(xy + xz) - (xy + z) = even - odd
xz - z = odd
z(x-1) = odd.
Since odd*odd = odd, x-1 must be odd, implying that x itself is even.
SUFFICIENT.
Statement 2: y + xz is odd
(y + xz) + (xy + z) = odd + odd
y + x(z+y) + z = even
x(y+z) + (y+z) = even
(y+z)(x+1) = even.
Since it's possible that x+1 is even or that x+1 is odd, we cannot determine whether x is even.
INSUFFICIENT.
The correct answer is
A.
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