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16 Sep 2014, 01:07
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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 must be odd?
A. \(A + D\)
B. \(B + D\)
C. \(C + D\)
D. \(A + B\)
E. \(A + C\)
A. \(A + D\)
B. \(B + D\)
C. \(C + D\)
D. \(A + B\)
E. \(A + C\)
16 Sep 2014, 01:07
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Official Solution:
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 must be odd?
A. \(A + D\)
B. \(B + D\)
C. \(C + D\)
D. \(A + B\)
E. \(A + C\)
Subtract the first equation from the second:
\((D + B - A) - (A - C + B) = odd - even\);
\(C + D - 2A = odd\);
\(C + D = odd + 2A\);
\(C + D = odd + even\);
\(C + D = odd\).
Answer: C
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 must be odd?
A. \(A + D\)
B. \(B + D\)
C. \(C + D\)
D. \(A + B\)
E. \(A + C\)
Subtract the first equation from the second:
\((D + B - A) - (A - C + B) = odd - even\);
\(C + D - 2A = odd\);
\(C + D = odd + 2A\);
\(C + D = odd + even\);
\(C + D = odd\).
Answer: C
19 Jan 2016, 06:25
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There is an easier way this question can be solved.
Lets say A-C+B = 2n (even)
and after rearranging the second equation, -A+D+B=2m+1 (odd)
Adding the two equations we get, 2B+D-C=2(m+n)+1
From this, 2B is even and 2(m+n) is even, hence we conclude,
D-C=odd
When difference of two integers is odd, there sum will also be odd;
Hence D+C = odd
Lets say A-C+B = 2n (even)
and after rearranging the second equation, -A+D+B=2m+1 (odd)
Adding the two equations we get, 2B+D-C=2(m+n)+1
From this, 2B is even and 2(m+n) is even, hence we conclude,
D-C=odd
When difference of two integers is odd, there sum will also be odd;
Hence D+C = odd
That solution isn't correct -- it could become correct, if the first value was "2k" and the second value was "2m + 1", but using "k" both times isn't right. That solution is a bit inefficient though. We know D + B - A is odd and A - C + B is even. So if we subtract that second expression from the first one, we must get something odd, so D + B - A - (A - C + B) is odd, and simplifying, D + C - 2A is odd. Subtracting the even number 2A won't change evenness or oddness at all, so D + C is odd. 
