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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? [#permalink]

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15 Aug 2017, 12:04
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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
[Reveal] Spoiler: OA

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

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15 Aug 2017, 12:22
Expert's post
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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

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

[Reveal] Spoiler:
D

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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?

$$x^2$$ - $$z^2$$ 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.

Quote:
(1) x + z is odd
(2) x - 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.

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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

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

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15 Aug 2017, 12:34
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
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