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
1. xy+xz is an even integer
2. y+xz is an odd integer
Target question: Is x an even integer?abhasjha wrote: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
Given: xy + z is an odd integer
Statement 1: xy + xz is an even integer
Notice that xy + xz and xy + z have an xy term in common.
Let's use the fact that EVEN - ODD = ODD
So, (xy + xz) - (xy + z) = ODD
Simplify: xz - z = ODD
Factor: z(x - 1) = ODD
IMPORTANT: If the product of two integer values is ODD, then BOTH values must be ODD.
So, z is ODD and (x - 1) is ODD.
If x - 1 is ODD, then x must be EVEN
Since we can answer the target question with certainty, statement 1 is SUFFICIENT
Statement 2: y + xz is an odd integer
Let's use the fact that ODD + ODD = EVEN
So, (y + xz) + (xy + z) = EVEN
Simplify: y + xz + xy + z = EVEN
Rearrange: xz + z + xy + y = EVEN
Factor in pieces: z(x + 1) + y(x + 1) = EVEN
Simplify: (z + y)(x + 1) = EVEN
At this point, we can see that (x + 1) can be either ODD or EVEN, which means x can be either ODD or EVEN
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT
Answer = A
Cheers,
Brent
