For integers x and y, if (x+2)(y+3) is even then 4xy must be divisible

Just looking at the question, (x+2)(y+3) could be even*even, odd*even, or even*odd in order for the whole expression to be even.

However, just looking at 4xy, since x*y will always be an integer, it will always be divisible by 4. 4 divided by 4 is 1, hence preserving x*y.

Answer: A. 4

ANSWER IS A. Notice WE are told that MUST NOT COULD .

AS we know that (X+2) (Y+3) Is even , X and Y NOT need to be even neccessairly . CONSIDER THIS case : Y and X could be ODD: If X is odd then X+2 is odd too and as

(X+2) (Y+3) is EVEN only one number is to be even and if Y is odd too then Y+3 Is even .so the whole expression is EVEN too.


SO we can have a case in which X and Y could be BOTH odd but the expression is even. in this case we would have 4*X*Y and ONLY is divisible BY 4 .

AS we are told That MUST BE DIVISIBLE ONLY Option A can be fit...

here is my explanation:
Given that (x+2)*(y+3) is even. therefore here are the possible cases:

(x+2) (y+3) (x+2) (y+3) (x+2) (y+3)
E E E O O E

x y x y x y
E O E E O O

xy xy xy
E E O

can be divided can be divided must only be
by 4, 8 etc... by 4, 8 etc... divided by 4

Hence A

Solving Algebraically.....

xy + 3x + 2y + 6 = 2k
xy = 2k - 3x - 2y - 6
4xy = 8k - 12x - 8y - 24....

Now with least value of x,y as 1,1 the condition of being even will suffice..

So only surity is 4xy divisible by 4.

VERITAS PREP OFFICIAL SOLUTION:

For this Even/Odd number property question, it's important to recognize that if either parenthetical term (x + 2) or (y + 3) is even, then the product of the two will be even, as Even * Odd or Even * Even is even.

This means that the given information could be held true if:

x is even

or

y is odd

If x is even, that means that 4xy would be 4(even)(y), which would allow that product 4xy to be divisible by 8 or more. But if x were odd and y were odd (for example, 1), then 4xy would be 4(odd)(odd), meaning that the only factors of 2 would come from the coefficient of 4. Therefore, as x = 1 and y = 1 are indeed possible, the only answer choice that MUST BE a factor of 4xy is 4.

Answer: A.

Answer = A= 4

Let x = 3 & y = 1

(x+2)(y+3) would be even with the above values

4xy = 12 would be divisible by 4

I didn't understand the question..
4XY will always be divisible by 4 because it already has 4 in it (already multiplied by 4)

why do we have to open any brackets and multiply?

Hi All,

DesiGmat has caught the 'secret' to this question. Unfortunately, the GMAT isn't likely to make it quite this "easy" on you, but there is something to be said for paying careful attention to what the specific prompt gives you and what the specific question asks for...

GMAT assassins aren't born, they're made,
Rich

hi desigmat ,

it is not so simple that you dont require to open brackets....
it was simple if no conditions were set as shown in red..
if (x+2)(y+3) is even has to be checked in different scenarios for evenness
in this case, you may get the answer without opening the brackets,here are two ways
1) what if it gives you,for eg, x and y are consecutive positive integers ....
this would ensure that one of x nd y will be even, so 4xy must be div by 8 in this case..
2) what if Q says if (x+2)(y+4) is even, and x,y are different integers..
here both x and y have to be even, so 4xy must be div by 8...

What i am trying to tell you is that you have to check a Q for all possibilities before hooking on to an ans

No need to solve this one
4xy must be divisible by 4xy for x and y are integers
Smash that A

Irrespective of the integer value of x & y, '4' * X*Y will always be divisible by 4.

Ans A

Solution

Given

• For integers x and y, (x+2)(y+3) is even.

To find

• 4xy must be divisible by which number?

Approach and Working out

• (x+2)(y+3) = Even

o Case 1 - ) x+2 = Even and y +3 = Odd

 x = Even and y= Odd

• 4xy will be definitely divisible by 8


o Case 2 - ) x+2 = Odd and y +3 = Even

 x = Odd and y= Odd

• 4xy will be definitely divisible by 4

In both the cases, 4xy is definitely divisible by 4.

Thus, option A is the correct answer.
Correct Answer: Option A

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