tricky odd/even integer question

[Baldini]

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

Could someone please explain how to get to the answer as quickly as possible?

Thanks

[bluementor]

There are 4 cases that will satisfy xy + z = odd
(i) x = even, y = even, z = odd
(ii) x = even, y = odd, z = odd
(iii) x = odd, y = even, z = odd
(iv) x = odd, y = odd, z = even

is x even?

Statement 1:

xy + xz = even
x(y + z) = even

case (a): x = odd, (y + z) = even. this is not possible since this will not satisfy any of the cases(i) - (iv).

case (b): x = even, (y + z) = odd/even. this satisfies cases (i) and (ii).

so, since only case b is valid, x must be even. sufficient.

Statement 2:

y + xz = odd

case (c): y = odd, xz = even. this satisfies cases (ii) and (iv) where x could either odd or even. insufficient.
case (d): y = even, xz = odd. this satisfies case (iii), where x is odd.

so, since this statement doesn't provide any restriction on x, it is not sufficient.

Choose A.

-BM-

[Baldini]

Thanks a lot Bluementor.
Can I just ask how long it took you to get the OA? Is there anything you did to streamline your process (took me a long time to do this)?
thanks

[Ian Stewart]

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

Could someone please explain how to get to the answer as quickly as possible?

Thanks

At least when considering S1, you can proceed quickly if you notice the similarity between the expression in S1 and the expression in the question. We know xy + xz is even, and that xy + z is odd; we can subtract to eliminate the xy:

xy + xz - (xy + z) = even - odd
xz - z = odd
z(x - 1) = odd

so z is odd, and x - 1 is odd, so x is even. S1 is sufficient.

[iamcste]

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

Could someone please explain how to get to the answer as quickly as possible?

Thanks

At least when considering S1, you can proceed quickly if you notice the similarity between the expression in S1 and the expression in the question. We know xy + xz is even, and that xy + z is odd; we can subtract to eliminate the xy:

xy + xz - (xy + z) = even - odd
xz - z = odd
z(x - 1) = odd

so z is odd, and x - 1 is odd, so x is even. S1 is sufficient.

Cool solution

[cubicle_bound_misfit]

if this is of any help

from stmt 1,

xy +z +(x-1)z == even

odd + (x-1)z == even

hence (x-1)z is odd

hence x-1 odd and z odd

hence x even and z odd

putting back in the original condition

xy = even
z =odd

which satisfies the condition hence yes x is even, choose A/D

for stmt 2,

my reasoning is same as 'Bluementor'

hence answer A.

regards,
CBM

[bblast]

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

Could someone please explain how to get to the answer as quickly as possible?

Thanks

At least when considering S1, you can proceed quickly if you notice the similarity between the expression in S1 and the expression in the question. We know xy + xz is even, and that xy + z is odd; we can subtract to eliminate the xy:

xy + xz - (xy + z) = even - odd
xz - z = odd
z(x - 1) = odd

so z is odd, and x - 1 is odd, so x is even. S1 is sufficient.

Hi Ian,
I remember you posted a similar solution to a similar GMAT club problem I posted here. Thanks for this approach on statement 1. Can we use similar method to deduce a result from statement 2 ? Or is testing cases is the only method out ?

[navami]

Good question thanks. considering option 2. was little difficult.

[czarczar]

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

Could someone please explain how to get to the answer as quickly as possible?

Thanks

At least when considering S1, you can proceed quickly if you notice the similarity between the expression in S1 and the expression in the question. We know xy + xz is even, and that xy + z is odd; we can subtract to eliminate the xy:

xy + xz - (xy + z) = even - odd
xz - z = odd
z(x - 1) = odd

so z is odd, and x - 1 is odd, so x is even. S1 is sufficient.

Perfect approach!

[prateek_guy2004]

Statement 1 sufficient


1. xy + xz is an even integer

x(y+z) = even

2. y + xz is an odd integer Statement 2 Not sufficient.

Answer A

[GMATGuruNY]

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:
Test whether it's possible that x is odd.
Let x=1.
Substituting x=1 into xy+z = ODD, we get:
y+z = ODD
Substituting x=1 into xy+xz = EVEN, we get:
y+z = EVEN
The equations in red contradict each other, implying that it is not possible that x is odd.
Thus, x must be even.
SUFFICIENT.

Statement 2:
Test whether it's possible that x is odd.
Let x=1.
Substituting x=1 into xy+z = ODD, we get:
y+z = ODD
Substituting x=1 into y+xz = ODD, we get:
y+z = ODD
Since the equations in green are the same, it's possible that x is odd.

Test whether it's possible that x is even.
Let x=2.
Substituting x=2 into xy+z = ODD, we get:
2y+z = ODD
Substituting x=2 into y+xz = ODD, we get:
y+2z = ODD
The equations in green are both viable if y=1 and z=1, implying that it's possible that x is even.

Since the answer to the question stem is NO in the first case but YES in the second case, INSUFFICIENT.

The correct answer is A.