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K and T

by anuptvm » Wed Dec 29, 2010 6:03 am
If K and T are integers and K^2 - T^2 is an ODD integer, which of the following must be an even integer?

I. K + T + 2

II. K^2 + 2KT + T^2

III. K^2 + T^2

My choice was II, OA is None.

Here is how I approached it. Given K^2 - T^2 is ODD => K is ODD and T is ODD

Checking II with ODD numbers gives ODD + EVEN + ODD => which is EVEN

Am I making a mistake?
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by Night reader » Wed Dec 29, 2010 6:21 am
anuptvm wrote:If K and T are integers and K^2 - T^2 is an ODD integer, which of the following must be an even integer?

I. K + T + 2

II. K^2 + 2KT + T^2

III. K^2 + T^2

My choice was II, OA is None.

Here is how I approached it. Given K^2 - T^2 is ODD => K is ODD and T is ODD

Checking II with ODD numbers gives ODD + EVEN + ODD => which is EVEN

Am I making a mistake?
Let's examine each case:
I. K + T + 2 => even+odd+even => odd
II. K^2 + 2KT + T^2 => (K+T)^2 => (even+odd)^2 OR odd^2 => odd
III. K^2 + T^2 => even^2 + odd^2 => odd

Answer: None
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by anshumishra » Wed Dec 29, 2010 6:22 am
anuptvm wrote:If K and T are integers and K^2 - T^2 is an ODD integer, which of the following must be an even integer?

I. K + T + 2

II. K^2 + 2KT + T^2

III. K^2 + T^2

My choice was II, OA is None.

Here is how I approached it. Given K^2 - T^2 is ODD => K is ODD and T is ODD

Checking II with ODD numbers gives ODD + EVEN + ODD => which is EVEN

Am I making a mistake?
K^2 - T^2 = (K+T)(K-T) = ODD
=> K+T -> odd and
K-T = odd

I.K+T+2 = Odd + 2 = Odd
II. K^2+2KT+T^2 = (K+T)^2 = Odd*Odd = odd
III.K^2+T^2 = (K+T)^2-2KT = Odd - Even = Odd

Hence OA should be None.
Thanks
Anshu

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by Anurag@Gurome » Wed Dec 29, 2010 6:38 am
anuptvm wrote:If K and T are integers and K^2 - T^2 is an ODD integer, which of the following must be an even integer?

I. K + T + 2
II. K^2 + 2KT + T^2
III. K^2 + T^2
.... (K² - T²) is odd
=> (K + T)(K - T) is odd
=> (K + T) and (K - T) both are odd

I. (K + T + 2) = (K + T) + 2 = ODD + 2 = ODD
II. (K² + 2KT + T²) = (K + T)² = (ODD)² = ODD
III. (K² + T²) = (K + T)² - 2KT = (ODD)² - EVEN = ODD - EVEN = ODD

Therefore none of them is even.
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by anshumishra » Wed Dec 29, 2010 6:42 am
Night reader wrote:
anuptvm wrote:If K and T are integers and K^2 - T^2 is an ODD integer, which of the following must be an even integer?

I. K + T + 2

II. K^2 + 2KT + T^2

III. K^2 + T^2

My choice was II, OA is None.

Here is how I approached it. Given K^2 - T^2 is ODD => K is ODD and T is ODD

Checking II with ODD numbers gives ODD + EVEN + ODD => which is EVEN

Am I making a mistake?
Let's examine each case:
I. K + T + 2 => even+odd+even => odd
II. K^2 + 2KT + T^2 => (K+T)^2 => (even+odd)^2 OR odd^2 => odd
III. K^2 + T^2 => even^2 + odd^2 => odd

Answer: None
Night reader,

I guess you made a small mistake here.
K and T can be either even or odd (Your solution relies on K being Even and T as Odd. For e.g. K can be 5, T can be 2, and still K^2 -T^2 = odd).
Thanks
Anshu

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by Night reader » Wed Dec 29, 2010 6:54 am
reply to previous @Anshu

i employ distributive rule here: even+odd=odd+even => k and t of course can be either even or odd, that follows from the question stimuli, which we had to analyze beforehand K^2 - T^2 is an ODD integer => only possible (k-t)(k+t) when one is even and one is odd
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by anshumishra » Wed Dec 29, 2010 7:00 am
Night reader wrote:reply to previous @Anshu

i employ distributive rule here: even+odd=odd+even => k and t of course can be either even or odd, that follows from the question stimuli, which we had to analyze beforehand K^2 - T^2 is an ODD integer => only possible (k-t)(k+t) when one is even and one is odd
Excellent !
So that essentially means Either one of K and T is even and the Other is Odd , so they are never even or odd at the same time. Perfect ! May be you should add these lines in your solution to make it clear. That is a nice approach.
Thanks
Anshu

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