If x and y are integers and x^2 + y^2 is even, which of the following

Question

If x and y are integers and x^2 + y^2 is even, which of the following may be odd?

I. xy + 1
II. 3x + 5y
III. xy + y

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

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aditya1818 · Answer

If x and y are integers and x^2 + y^2 is even, which of the following may be odd?

I. xy + 1
II. 3x + 5y
III. xy + y

For x^2 + y^2 is to be even x^2 and y^2 should be even or both should be odd simultaneously and thus x and y are either even or odd simultaneously. So x and y =Odd or x and y= Even

Lets now evaluate Statements:
I. xy + 1: x and y both even thus xy is even and (xy+1) is odd. so statement I can be odd.

II. 3x + 5y : if x and y both are even then 3x and 5x both are even then 3x+5x is also even
If x and y both are odd then 3x and 5x both are odd and Odd + Odd = Even so in any case Statement II will not fetch odd number. Statement II is not correct.

III. xy + y
if x and y both are even then xy + y gives us even result
if x and y both are odd then xy + y gives us even results 
Statement III is also incorrect.

Only Statement I is correct thus Option (A) is the winner.

007Phoenix007 · Answer

The expression x2+y2 is even meaning either x and y both are even or odd

Case 1: x and y both even
 x=2a y=2b
I: xy+1= 4ab+1, which can be odd for some intergers.

II: 3x+5y = 6a+10b = 2(3a+5b), which is even in all values

III: xy+y = 4ab+2b = 2(2ab+b), which is again even in every value

Case 2: x and y both odd
 x= 2a+1 y= 2b+1

I: xy+1 = (2a + 1)(2b +1) = 4ab+2a+2b+2, which is even but as case 1 is odd then this MAY be odd

II: 3x+5y = 3(2a+1)+5(2b+1) = 6a+10b+8, which always even

III: xy + y = y(x+1), where x is odd so x+1 is even, so tis is always even

Conclusion 
Expression II & III are always even but Expression I can be odd some some values

So only I can be odd 
Therefore option A

ravi1522 · Answer

Hello ,
If x and y are integers and x^2 + y^2 is even, which of the following may be odd?

now as per equation either x and y will be odd or both will be even .

I. xy + 1
if both are even then even plus odd can be odd 
this equation can be correct 
II. 3x + 5y
if x and y are even then above equation will even and if both are odd then odd+ odd is again even 
so this equation cannot be odd
incorrect 
III. xy + y
if xand y are odd then above equation will be odd +odd =even 
if both are even then also even + even again even 
This cannot be odd

so only equation 1 is correct

Answer is Option A

rohitz98 · Answer

Given x^2 + y^2 is even which means x,y should both be either odd or even
Taking first choice
i) xy + 1 If x, y is even then it might be odd
ii) 3x + 5y If x,y both are odd or even it'll be even
iii) xy + y => x(y+1) Here it'll always be even.

Hence  Option A  is correct answer

Dereno · Answer

X and Y are integers, means they can be both positive or negative.

x^2 + y^2 is even.

Sum of squares of two integers is Even, only when two cases happen.

Case 1: Odd + Odd = Even

Case 2: Even + Even = Even

which of the following   may be odd  ?
 
I. xy + 1
II. 3x + 5y
III. xy + y

I. xy + 1

Applying case 1 and Case 2, we get

odd * odd + 1 = odd + 1 = Even

even * even + 1 = even + 1 = Odd

I may be ODD.

II. 3x + 5y

Applying case 1 and Case 2, we get

3*odd + 5* Odd = Odd + Odd = Even

3* Even + 5* Even = Even + Even = Even

II cannot be odd.

III. xy + y

Applying case 1 and Case 2, we get

Odd * Odd + Odd = Odd + Odd = Even

Even * Even + Even = Even + Even = Even

III cannot be odd.

Option A

AsifAhmadRahmani · Answer

This is an interesting question here is the question asking which must be ODD.
The fastest way to solve it is the application of the formula of Even and Odd but also beware of the trap.

Lets breakdown the question stem-

If x and y are integers and x^2 + y^2 is even, which of the following may be odd?

I. xy + 1
II. 3x + 5y
III. xy + y

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

This Daily Butler Question was provided by Experts' Global

If x and y are integers and x^2 + y^2 is even, which of the following

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GMAT Problem Solving (PS) Questions

If x and y are integers and x^2 + y^2 is even, which of the following

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