If x and y are integers and x y is odd, which of the following must

Question

If x and y are integers and x − y is odd, which of the following must be true?

I. xy is even.
II. x^2 + y^2 is odd.
II. (x + y)^2 is even.

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

(PS01120)

KSBGC · Accepted Answer

Given

x - y = odd

In this case either x or y must be even and another must be odd.

I. xy = even. True . (3*2)= 6

II. x^2 + y^2 = odd . True.  (2)^2 + (3)^2 = 13

III.  (3+2)^2 = 25  . False.

The best answer is D.

bhanu88 · Answer

D...odd-even or even-odd is odd...
I. One is even so xy will be even
II. Odd+even is odd/ even +odd is odd
III... Odd+even is odd ...sqaure can't be even ..will be odd

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dave13 · Answer

IMO D must be correct answer

Let X = 7 and Y = 2 I. xy is even ( 7*2 = 14 ) correct

II. x^2 + y^2 is odd. 7^2+2^2 = 53 TRUE II. (x + y)^2 is even. (7+ 2)^2 = 49+4 =53 FALSE EDIT

ScottTargetTestPrep · Answer

The only way to get an odd sum or difference of two integers is if one integer is odd and the other integer is even. Since x - y is odd, then x is even and y is odd OR x is odd and y is even.

We recall that even x odd = even. Thus, xy must be even. Thus, Roman numeral I is true.

Since odd + even = odd, we see that x^2 + y^2 must be odd, and (x + y)^2 must be odd. Thus, Roman numeral II is true and Roman numeral III is false.

Answer: D

Salsanousi · Answer

Can be solved much faster and without any substitution if we know that even - odd = odd. From this we know that one number must be even. What applies to subtraction in terms of even/odd applies to addition.

even * odd = even

Having a power does not change whether the number is odd or even.

only I and II are true.

TheGuyDangerous · Answer

Input numbers 3&2 and 5&2.

Smash D

EgmatQuantExpert · Answer

Given  
 • x and y are integers and x − y is odd
 o One of x and y is even and another is odd.

To find  
 • The correct option.

Approach and Working out

I – xy is even
 • Multiplication of one even and one integer is always even.

II - x^2 + y^2 is odd.
 • Even ^2 + Odd^2 = Even + Odd = Odd

III - (x + y)^2 is even.
 • Odd^2 = Odd
• Not true

Thus, option D is the correct answer.

Correct Answer: Option D

perryzhu · Answer

Hi Payal

why don't we need to consider 0 here? it is an integer right?

3-0=3 but 3x0=0 is not even.

Basshead · Answer

If x and y are integers and x − y is odd, which of the following must be true?

For x - y to be odd, x and y must be odd/even or even/odd.

I.  xy  is even. -- MUST BE TRUE
II.  x^2 + y^2  is odd. -- MUST BE TRUE
II.  (x + y)^2  is even. -- FALSE

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

Answer is D.

LostGMATFellow · Answer

Asking the same question as perryzhu. Why isnt 0 considered here?

GmatKnightTutor · Answer

0 is considered even from a mathematical point of view. "3x0=0" would be even therefore.

Bunuel · Answer

Zero is an EVEN integer.

ZERO:

1. Zero is an  INTEGER .

2. Zero is an  EVEN  integer.

3. Zero is  neither positive nor negative  (the only one of this kind)

4. Zero is  divisible by EVERY integer except 0 itself  ( \frac{x}{0} = 0 , so 0 is a divisible by every number, x).

5. Zero is a  multiple of EVERY integer  ( x*0 = 0 , so 0 is a multiple of any number, x)

6. Zero is  NOT a prime number  (neither is 1 by the way; the smallest prime number is 2).

7. Division by zero is NOT allowed:  anything/0 is undefined .

8. Any non-zero number  to the power of 0 equals 1  ( x^0 = 1 )

9.  0^0  case is  NOT tested on the GMAT .

10. If the exponent n is positive (n > 0),  0^n = 0 .

11. If the exponent n is negative (n < 0),  0^n  is undefined, because  0^{negative}=0^n=\frac{1}{0^{(-n)}} = \frac{1}{0} , which is undefined.  You CANNOT take 0 to the negative power .

12.  0! = 1! = 1 .

2. Properties of Integers  
 
    Theory    
 
 Math: Number Theory 
 Number Properties: Tips and hints 
 Advanced Number Properties on GMAT (All 5 parts) 
 Do you make these 3 mistakes in GMAT Even-Odd Questions?  
 Everything about Factorials on the GMAT 
 Solving Complex Problems Using Number Line 
 
    Questions    
 
 Tough New Set: Number Properties Set 
 Power of a Number in a Factorial Problems 
 Number Properties Ability Quiz 
 Special Numbers and Sequences Problems 
 The E-GMAT Number Properties Knockout: 10 Great 700+ Ques for you! 
 Trailing Zeros Problems 
 DS Questions 
 PS Questions

For more check  Ultimate GMAT Quantitative Megathread

Hope it helps.

If x and y are integers and x y is odd, which of the following must

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If x and y are integers and x y is odd, which of the following must

Prep Toolkit

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