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

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

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 cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Re: If x and z are positive integers, is x^2 - z^2 odd? [#permalink]

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15 Aug 2017, 12:25

Skywalker18 wrote:

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

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

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
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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.
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Re: If x and z are positive integers, is x^2 - z^2 odd?
[#permalink]
15 Aug 2017, 12:34