Problem Solving

If x and y are positive integers, is the product xy even?

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

5x - 4y = even (if 5x and 4y are even or both are odd)
4y can never be odd (any number multiply with even is even) so 5x and 4y both odd are ruled out.
5x = > even if X is even
if X is even then product of xy is even

A alone suffucient

same applys to 6x + 7y B alone is sufficient

IMO D

Ok thanks to Stuart Kovinsky
He once said that
if either of the numbers are even then product is even....
so here, what you have to see is,
find both odd numbers condition that satisfies the equation
ie
5x-4y is even

if x=1 and y=3 OR x=1 and y=1......
but none of these satisfies things...
same for 2
so we have answer in both situations
so D

The key to this is a good rephrase. For a product of 2 integers to be even, at least one of them must be even.

REPHRASE: Is x or y even?

The statements should be evaluated with even/odd in mind. For instance look at (1). 4y is always even. so 5x-4y is 5x-even. For this to have an even result, 5x itself must be even, so x must be even. This answers our rephrase with a definitive YES.

The 2nd statement should be handled the same way. Again, think in terms of even/odd property. The answer is D

This is GMATPrep question 1510. To practice similar questions, set topic='Number Properties' and difficulty='400-500' in the Drill Generator

-Patrick

review the following:

even*even=even
even*odd=even
odd*odd=odd

therefore, for x*y to be even, x has to be even/odd and y has to be even or x has to be even and y has to be even/odd

from statement 1:
5x-4y= even
so x is even and y is even/odd
(if x is odd then equation is odd)
so even*even or even*odd= even
sufficient

from statement 2:
6x+7y=even
so x is even/odd and y is even
(if y is odd then sum=odd)
so even*even=even or odd*even=even
sufficient

d

Only case when xy will not be even will be when both x and y are odd. Let us see if we can find this information from the given statements.

(1) says that 5x - 4y is even
Now, even - even is even and odd - odd is also even.
Case 1: Both 5x and 4y are even
=> x is even (because only then 5x will be even) and y can be even or odd
=> Since x is even, xy will be even.

Case 2: Both 5x and 4y are odd
This case is not possible because 4y can never be odd.

So, we know from (1) that xy will be even

Sufficient.


(2) says 6x + 7y is even
Now, even + even is even and odd + odd is also even.

Case 1: Both 6x and 7y are even
=> x can be even or odd, and y is even (because only then 7y will be even)
=> Since y is even, xy will be even.

Case 2: Both 6x and 7y are odd
This case is not possible because 6x can never be odd.

So, we know from (2) that xy will be even

Sufficient.