If x and y are positive integers...

[jasonmaehara]

If X and Y are positive integers, is the product XY even?

1) 5x - 4y is even
2) 6x + 7y is even

Answer: D

Could use some help as to why! Thanks

[smohit04]

Rephrasing - we need to find if either X or Y is even..?

(1)- 5x - 4y is even

Since 4y is even (irrespective of y being odd/even), 5x has to be even. Thus x will be even - SUFFICIENT

(2) - 6x + 7y is even

Since 6x is even (irrespective of x being odd/even), 7y has to be even. Thus y will be even - SUFFICIENT

[RonPurewal]

Rephrasing - we need to find if either X or Y is even..?

(1)- 5x - 4y is even

Since 4y is even (irrespective of y being odd/even), 5x has to be even. Thus x will be even - SUFFICIENT

(2) - 6x + 7y is even

Since 6x is even (irrespective of x being odd/even), 7y has to be even. Thus y will be even - SUFFICIENT

excellent.

--

the other way to do these problems, if you get stuck getting through the kind of reasoning given above, is simply to slog through all of the different possibilities for whether x and y are even or odd, and just see which ones work.

here:
(1) 5x - 4y is even

4 possibilities for (x, y):
(odd, odd): 5x - 4y = odd - even = odd; this doesn't work.
(odd, even): 5x - 4y = odd - even = odd; this doesn't work.
(even, odd): 5x - 4y = even - even = even; this works.
(even, even): 5x - 4y = even - even = even; this works.
in both of the cases that work, x is even, so xy is also even.
SUFFICIENT


(2) 6x + 7y is even

4 possibilities for (x, y):
(odd, odd): 6x + 7y = even + odd = odd; this doesn't work.
(odd, even): 6x + 7y = even + even = even; this works.
(even, odd): 6x + 7y = even + odd = odd; this doesn't work.
(even, even): 6x + 7y = even + even = even; this works.
in both of the cases that work, y is even, so xy is also even.
SUFFICIENT

[jasonmaehara]

Got it, thanks!

[tim]

glad to hear it!