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

Question

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

GSBae · Accepted Answer

1)  4y  will always be even. Then we have  5x - even = even . For this to be the case,  5x  must be even. Since 5 can't be even, then x must be even. Thus the product  xy  will be even.   Sufficient.

2)  6x  will always be even. Then we have  even + 7y = even . Thus  7y  is even, and  y  is even, and  xy  is even.  Sufficient.

Answer: D

GMAT TIGER · Answer

D. in either case x and y has to be even.

(1) 5x - 4y is even: x has to be even. suff.
(2) 6x + 7y is even: y has to be even. suff.

x2suresh · Answer

agree with D.

1) 5x - 4y is even -> x must be multiple of 2
 5*2k-4y = 2*(any number) even
sufficient
2)6x + 7y is even[/
y should multiple of 2

even
sufficient

BrentGMATPrepNow · Answer

Target question :    Is xy even?

Statement 1 : 5x - 4y is even   
Let's test all 4 cases 
 case a : x is even and y is even: In this case 5x-4y is  EVEN 
 case b : x is even and y is odd: In this case 5x-4y is  EVEN 
 case c : x is odd and y is even: In this case 5x-4y is ODD
 case d : x is odd and y is odd: In this case 5x-4y is ODD
So, cases a and b are both possible. 
In both cases the product xy is even
So,  xy  must  be even 
Since we can answer the  target question  with certainty, statement 1 is SUFFICIENT

Aside: If anyone is interested, we have a video (below) on testing possible cases for these question types

Statement 2 : 6x+7y is even 
Let's test all 4 cases 
 case a : x is even and y is even: In this case 6x+7y is  EVEN 
 case b : x is even and y is odd: In this case 6x+7y is ODD
 case c : x is odd and y is even: In this case 6x+7y is  EVEN 
 case d : x is odd and y is odd: In this case 6x+7y is ODD
So, cases a and c are both possible. 
In both cases the product xy is even
So,  xy  must  be even 
Since we can answer the  target question  with certainty, statement 2 is SUFFICIENT

Answer = D

Divyadisha · Answer

For xy to be even, either x or y or both must be even.

(1) 5x - 4y is even
E-E=E
O-O=E

Since 4y is even, 5x has to be even. 5 is not even, hence x is even.

xy will be even.

(2) 6x + 7y is even
E+E= E
O+O=E

Since 6x is even, 7y has to be even. And because 7 is not even y has to be even.

xy will be even.

Both statements are sufficient.

D is the answer

grimm1111 · Answer

In my opinion, this type of problem is super easy if you really understand what you're being asked, and then re-frame the question to be more straightforward. It's more about the question than the math.

Given: (x,y) > 0, AND are integers

Original Question:  Is xy even?

We know that any number multiplied by an even number is even, so the question is really asking:  Do we know if either "X" or "Y" (or both, or neither) is even?

Reframing the question this way is better, in my opinion.

A.) 5x - 4y is even.

For 5x - 4y to be even, both 5x and 4y have to be  of the same parity.  ("Parity" is the word that describes whether a number is odd or even)
In other words, odd + odd, or even + even, produce even results.

5x -  4y

We know off the bat that "4y" is even, because 4 is even. 4 times anything would be even. So we have  (  5x  - EVEN NUMBER = EVEN NUMBER. )  Therefore, we know that "5x" is also even, because 4y (even) and 5x have to be the same parity. (even minus even = even)

In order to make 5x even,  "x" has to be even.  That's because 5 is odd.

Knowing that "x" is even is enough to answer the question: Do we know if either "X" or "Y" (or both, or neither) is even?

SUFFICIENT.

B.) 6x + 7y is even

Same drill here. For 6x + 7y to be even, both terms have to be the same parity. Since 6 is even, we know that 6x is even. Since 6x is even, we know that 7y is even. Since 7y is even,  we know that y is even  (since 7 is odd).

Knowing that "y" is even is enough to answer the question: Do we know if either "X" or "Y" (or both, or neither) is even?

SUFFICIENT

The answer is D

susheelh · Answer

S1: 5x - 4y = even => 5x = even+4y => 5x = even => x = even => xy = even. Suff

S2: 6x+7y = even => 7y = even - 6x =>7y = even => y = even => xy = even. Suff

Both S1 and S2 are individually sufficient. Final answer = D

merajul · Answer

If x and y are positive integers, is the product xy even
(1) 5x - 4y is even
(2) 6x + 7y is even

1. for 5x-4y = Even, x must be even & y may or may not be even. Since x is even, we can conclude xy is even. SUFFICIENT.
 Even – Even = Even 
Even – Odd = Odd
Odd – Even = Odd
Odd – Odd = Odd

2. for 6x + 7y is even, y must be even. Since y is even, we can conclude xy is even. SUFFICIENT.
 Even + Even = Even 
Odd + Even = Odd
Even + Odd = Odd

masoomjnegi · Answer

Statement 1. 5x – 4y is even i.e. 5x – 4y = 2k
Difference of two numbers is even if either both of them are odd or both of them are even.
Here, 4y is always even. So, 5x is also even.
Since 5x is even, we can say that x is even.
Now, xy will be even because the product of an even no with any positive integer is always even. Hence, Sufficient.
Statement 2. 6x + 7y is even.
6x is eve n because it is divisible by 2. So, 7y has to be even.
For 7y to be even, y has to be even.
We know the product of an even no with a positive integer is even.
Hence, xy is even. Hence, Sufficient.

ed1deee · Answer

Can this problem be broken out algebraically rather than testing values?

e.g.

For Statement 1 I did: 
(1) 5x - 4y is even

4x+1x - 4y = E
4x -4y +1x = E
4(x-y) + 1x = E; X MUST BE EVEN; if X is even XY = Even

so sufficient;

(2) 6x + 7y is even
6x + 6y + y = Even
6(x+y) + y = Even; Y must be even as E + E = 0 and 6*(X+Y) = E

so sufficient;

Answer = D

GMATWhizTeam · Answer

Step 1: Analyse Question Stem

It is known that x and y are positive integers. Therefore, each of x and y are either odd or even.
We have to find out if the product of x and y is even.

The product of two numbers is even if at least one of them is even.

Step 2: Analyse Statements Independently (And eliminate options) – AD / BCE

Statement 1:  5x - 4y is even

Since y is an integer, it can be said that the term 4y is definitely even.

Therefore, 5x – even = even

Or 5x = even + even = even

5x can be even only when x is even. 
If x is even, the product xy is definitely even.

The data in statement 1 is sufficient to answer the question with a definite YES.
Statement 1 alone is sufficient. Answer options B, C and E can be eliminated.

Statement 2:  6x + 7y is even

Since x is an integer, it can be said that the term 6x is definitely even.

Therefore, even + 7y = even.

Or 7y = even – even = even

7y can be even only when y is even.
If y is even, the product xy is definitely even.

The data in statement 2 is sufficient to answer the question with a definite YES.
Statement 2 alone is sufficient. Answer option A can be eliminated.

The correct answer option is D.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards