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If A, B, C, and D are integers such that A - C + B is even [#permalink ]
20 Sep 2011, 02:29

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If A, B, C, and D are integers such that A - C + B is even and D + B - A is odd, which of the following expressions is always odd?

A. A + D

B. B + D

C. C + D

D. A + B

E. A + C

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Last edited by

Bunuel on 05 Mar 2013, 02:02, edited 1 time in total.

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jamifahad wrote:

If A , B , C , and D are integers such that A - C + B is even and D + B - A is odd, which of the following expressions is always odd? A + D B + D C + D A + B A + C I am wondering if there is any matrix method to do these sort of Even/Odd question. If anybody knows about any such method, kindly post.

***THIS SOLUTION HAS FLAWS AS POINTED OUT BY TomB*** Sorry, didn't use the matrix for this one.

A-C+B=even;

means

Either one of these

Or two of these

Or All three of them

ARE EVEN.

But, NOT all of them are ODD.

D + B - A=odd

Means, ALL of them are ODD

That confirms; A, B, D are ODD; and from before, C must be EVEN.

Ans: "C"

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Last edited by

fluke on 21 Sep 2011, 08:46, edited 1 time in total.

Added the warning after TomB pointed out error

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Hi Jamifahad, Even matrix method takes more than two mins in this problem since you have to jot down the even and odd values. Try subtracting the equations (A - C + B) - (D + B - A) = 2A-(C+D) Even - Odd = Odd So 2A-(C+D) is odd 2A is always even , so the subtrahend should always be odd, so C+D is odd. Hope I am clear.

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@abhi6001

Awesome buddy!

+1. and welcome.

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Hai fluke How did you get that D,B,A are odd. My reasoning is as follows: D + B- A=odd D=odd and B-A could be even . odd+even =odd. B-A= odd-odd=even;even-even=even. Then D is odd but A,B could be even or odd. another possibility: D + B- A=odd. so D= even and B-A=odd, then eithe B or A could be either even or odd. Therefore D=even, B and A are either even or odd. please correct me if i am wrong

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hai @abhi6001 can you explain 2A-(C+D)=odd. how C+D is odd. Thanks

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TomB wrote:

Hai fluke How did you get that D,B,A are odd. My reasoning is as follows: D + B- A=odd D=odd and B-A could be even . odd+even =odd. B-A= odd-odd=even;even-even=even. Then D is odd but A,B could be even or odd. another possibility: D + B- A=odd. so D= even and B-A=odd, then eithe B or A could be either even or odd. Therefore D=even, B and A are either even or odd. please correct me if i am wrong

My earlier post is wrong. thanks for pointing out. I'll have to use the matrix after all.

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jamifahad wrote:

If A , B , C , and D are integers such that A - C + B is even and D + B - A is odd, which of the following expressions is always odd? A + D B + D C + D A + B A + C

Start with all possible combinations for A and B:

ODD: O

EVEN: E

Now fill in the column 1, i.e. C's type with either odd or even; based on statement 1

Now fill in the column 4, i.e. D's type with either odd or even; based on statement 2

By POE, for each of the rows above:

"C+D=ODD"

Ans: "C"

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C. Even - Odd = Odd Therefore: A-C+B - (D+B-A) = ODD .. 2A-D-C = ODD ( now we know anything multiplied by 2 makes it Even.. so 2A cannot be the factor to cause this to be odd.. ) so we can simplfy it : -(D+C) = ODD Odd +/1 is ODD.. so D+C is odd!

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We know Even + Odd = Odd So, A - C + B = Even E - E + E = Even - 1 O - O + O = Even - 2 D + B - A = ODD 0 + E - E = ODD - 1 E + O - O = ODD - 2 So considering 1 and 2 If C is Even then D is odd & if C is Odd then D is Even So C + D should always be odd

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A-C+B = Even

D+B-A = Odd

is equivalent to

A+B+C = Even

A+B+D = Odd

We are only interested in C and D now, and need only to consider two scenarios: A+B is odd or even.

A+B C D Odd Odd Even Even Even Odd

C+D is always odd.

c)

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jamifahad wrote:

If A , B , C , and D are integers such that A - C + B is even and D + B - A is odd, which of the following expressions is always odd? A + D B + D C + D A + B A + C I am wondering if there is any matrix method to do these sort of Even/Odd question. If anybody knows about any such method, kindly post.

Looks good question....

We always need simple, easy and quick/fast approach:

A-C+B = EVEN ............I

D+B-A = ODD..............II

Add I and II:

A-C+B+D+B-A = EVEN+ODD

2B-C+D = ODD

No matter whether B is even or odd, 2B has to be an even number. If so, then remaining number, (D-C), has to be odd.

If (D-C) is odd, (D+C) has also to be always odd.

Therefore C.

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subtracting and adding the two terms

2A- (C+D) = odd meaning C+D = odd

2B - C+D = odd meaning (C-D) = odd. not provided in the options.

hence C it is.

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jamifahad wrote:

If A , B , C , and D are integers such that A - C + B is even and D + B - A is odd, which of the following expressions is always odd? A + D B + D C + D A + B A + C I am wondering if there is any matrix method to do these sort of Even/Odd question. If anybody knows about any such method, kindly post.

Make it plain and simple:

A - C + B = even

D + B - A = odd

Add both:

A - C + B + D + B - A = Even +odd

2B - C + D = odd

2B + (D - C) = odd

Since 2B is even, (D-C) must be odd. Alternatively, C could be odd or even and same is D but if C is odd, D must be even and vice versa. However it is not necessary to find out whether C is odd or even.

If so, either (C+D) or (C-D) is odd. So its C that

C + D is always odd.

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Re: If A, B, C, and D are integers such that A - C + B is even [#permalink ]
05 Mar 2013, 01:47
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Bunuel /Karishma/ Experts,

This is how I solved:

A−C+B is even and D+B−A is odd

Add them

2B + D-C must be odd

since 2B is even , D-C must be odd.

So C+D must be ODD :D

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Re: If A, B, C, and D are integers such that A - C + B is even [#permalink ]
05 Mar 2013, 02:06

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Re: If A, B, C, and D are integers such that A - C + B is even [#permalink ]
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Alternatively, you could also subtract both equations:

A- C + B = EVEN

D + B - A = ODD

Subtracting the 1st from the 2nd:

D +C - 2A = EVEN + ODD

Or,

D + C = EVEN + EVEN + ODD = ODD

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Re: If A, B, C, and D are integers such that A - C + B is even [#permalink ]
05 Mar 2013, 05:14

caioguima wrote:

Alternatively, you could also subtract both equations: A- C + B = EVEN D + B - A = ODD Subtracting the 1st from the 2nd: D +C - 2A = EVEN + ODD Or, D + C = EVEN + EVEN + ODD = ODD

yeah you can do that since these are equations.

but know that u shud never subtract in case of inequalities.. .

Kudos if this helps.

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Re: If A, B, C, and D are integers such that A - C + B is even [#permalink ]
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