If x and z are positive integers, is x^2 - z^2 odd?

Question

If x and z are positive integers, is  x^2-z^2  odd?

(1) x + z is odd
(2) x - z is odd

BrentGMATPrepNow · Answer

Great question!!

Target question :    Is x² - z² odd?

Given : x and z are positive integers

IMPORTANT RULES;
Rule #1) (ODD)² = ODD
Rule #2) (EVEN)² = EVEN
Rule #3) ODD + EVEN = ODD
Rule #4) EVEN + ODD = ODD
Rule #5) ODD - EVEN = ODD
Rule #6) EVEN - ODD = ODD

Statement 1 : x + z is odd   
By Rule #3 or Rule #4, we can conclude that one number (x or z) is ODD and the other number is EVEN. 
So, there are two possible cases: 
case a: x is ODD and z is EVEN
case b) x is EVEN and z is ODD

Let's examine each case. 
Case a: if x is ODD and z is EVEN, then x² is ODD and z² is EVEN. In this case,  x² - z² = ODD - EVEN = ODD 
Case b: if x is EVEN and z is ODD, then x² is EVEN and z² is ODD . In this case,  x² - z² = EVEN - ODD = ODD 
In either case,  x² - z² is ODD 
Since we can answer the  target question  with certainty, statement 1 is SUFFICIENT

Statement 2 : x - z is odd  
By Rule #5 or Rule #6, we can conclude that one number (x or z) is ODD and the other number is EVEN. 
So, there are two possible cases: 
case a: x is ODD and z is EVEN
case b) x is EVEN and z is ODD

Let's examine each case. 
Case a: if x is ODD and z is EVEN, then x² is ODD and z² is EVEN. In this case,  x² - z² = ODD - EVEN = ODD 
Case b: if x is EVEN and z is ODD, then x² is EVEN and z² is ODD . In this case,  x² - z² = EVEN - ODD = ODD 
In either case,  x² - z² is ODD 
Since we can answer the  target question  with certainty, statement 2 is SUFFICIENT

Answer: D

akshayk · Answer

x^2  -  z^2  can only be odd, if x and z are opposites i.e. if x is odd, z is even and if x is even, z is odd.

1) x + z is odd
=> one of them is odd, and one of them is even.
Sufficient.

2) x - z is odd
=> one of them is odd, and one of them is even.
Sufficient.

D is the answer.

gmatexam439 · Answer

My Answer - D

x^2-z^2  will be odd if either x or z will be add and other one will be even.

1. sum of an odd and an even number is add; rest all the times its even. Thus either of x or z is odd and the other is even. SUFFICIENT
2. subtraction of an odd and an even number is add; rest all the times its even. Thus either of x or z is odd and the other is even. SUFFICIENT

IanStewart · Answer

This is identical to a question in one of my books, so I'm curious where you found it. It's not a very original or unique setup, so I wouldn't be surprised if someone else designed it independently.

mimajit · Answer

we will solve this via the formula for ( A^2 - B^2 ) = ( A +B ) (A-B )
THEREFORE x^2-z^2 =( x+z) (x-Z)

1) x + z is odd
=> x +z is odd SO x -z IS ALSO is ODD.
eG : 4+3 = 7 , THEN 4-3 = 1 ( bOTH CASES ODD )

tHEN x^2-z^2 IS ALSO odd
Sufficient.

2) x - z is odd
=> x - z is odd, SO x +z IS ALSO ODD 
AS eG : 4+3 = 7 , THEN 4-3 = 1 ( bOTH CASES ODD ).

tHEN x^2-z^2 IS ALSO odd
Sufficient.

D is the answer.

bumpbot · Answer

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If x and z are positive integers, is x^2 - z^2 odd?

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If x and z are positive integers, is x^2 - z^2 odd?

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