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If A, B, C, and D are integers such that A - C + B is even [#permalink]
 20 Sep 2011, 03:29
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28% (01:40) wrong based on 2 sessions
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, 03:02, edited 1 time in total.
Edited the question
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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, 09:46, 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, 02:47
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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   Kudos if this helps anybody
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Re: If A, B, C, and D are integers such that A - C + B is even [#permalink]
05 Mar 2013, 03:06
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Re: If A, B, C, and D are integers such that A - C + B is even [#permalink]
05 Mar 2013, 06:08
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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, 06:14
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
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05 Mar 2013, 06:14
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