M20-04

Question

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

Bunuel · Accepted Answer

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

Abhigyanashakuntalam · Answer

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

bartone89 · Answer

I think I may have found a much more simple methodology to answer this question:

(1) A - C + B = even; thus, A - C + B = 2k (where k is an integer)

(2) D + B - A = odd; thus, D + B - A = 2k + 1 (where k is an integer)

rearrange (2): D + B - A - 1 = 2k

Sub: rearranged (2) into (1): A - C + B = D + B - A - 1 --> 2A + 1 = C + D

thus, per my method above, if C + D = 2A + 1 (where A is an integer), C + D is ALWAYS ODD

This solution is NOT correct. Check the discussion below for more.

sanket1991 · Answer

I think this question is  good  and  helpful .

passportsizephoto · Answer

bartone89  I believe your method is wrong because it assumes the values of the two equations are consecutive. ie 2k and 2k+1. I hope I havent misunderstood you solution.

Anyway, I think I may have found a better method.

The two equations are A-C+B and D+B-A, even and odd respectively.

On subtracting the two, you get, A-C+B-D-B+A = 2A-C-D (which is odd, since even - odd = odd)

2A-(C+D) is odd. Which means C+D is odd since we know that 2A has to be even.

C+D is one of the options. 
Hope this makes sense.

mvictor · Answer

I solved it in a different way:
add both expressions
A-C+B+D+B-A=odd.
ok, rewrite everything as:
2B+D-C=odd
2B is always even, it must mean only 1 thing: D-C is always odd. if D-C is odd, then D+C is always odd.

Raunak_Bhatia · Answer

Add or Substract the two equations-

A-C+B=Even
D+B-A=Odd
=2A+C-D (Even-Odd is always Odd) and we know 2A is always even

Therefore C-D should be Odd and thus also C+D

joondez · Answer

I agree very good solution. But it still requires timing luck or incredible number sense. I solved it through plugging in values. My method take no number sense needed and does not require too much logic.

A-C+B is even. Just plug in some easy numbers so that is true. A = 1 C = 2 B = 3 means that 1-2+3 = 2 which is even.
D+B-A is odd. Re-use the same variables from before. D + 3 - 1 is D + 2. Set D to an easy odd number which is 5. So D = 5. 
Now you have values for A through D or (A,B,C,D) = (1,3,2,5). Work through the answers.

A. A+D = 1 + 5 = 6
B. B+D = 3 + 5 = 8
C. C+D = 2 + 5 = 7. This is odd. No need to do the rest, the answer is C.
D. A+B
E. A+C

happypuppy · Answer

How can we say, 2k and 2k+1, or (2) - (1) = 1. We don't know about the value of A, B, C or D.
Rearrangement does not seem correct using 2k or 2k + 1 values.

Can you explain if I am wrong or missing something?

IanStewart · Answer

No, you're not missing anything

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.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

BottomJee · Answer

I think this is a  high-quality  question and I agree with explanation.

leepcoast · Answer

Joondez, actually it IS necessary to try the other alternatives, because according to the numbers you picked, option E would also be correct ( A + C = 1 + 2 = 3, which is odd)

AK760 · Answer

I think this is a  high-quality  question and I agree with explanation.

lnyngayan · Answer

Could you explain more on this?

(D + B - A) - (A - C + B) = odd - even ;

C + D - 2A = odd ;

Thankyou Bunuel!

Bunuel · Answer

D + B - A is odd
A - C + B is even

When we sutract we get:

(D + B - A) - (A - C + B) = odd - even

(D + B - A) - (A - C + B) gives C + D - 2A and odd - even is odd, so:

C + D - 2A = odd .

Manvi01 · Answer

I like the solution - it’s helpful.

saynchalk · Answer

in both equations, C and D are the reason the equation becomes even or odd, hence they must be even and odd (doesn't matter which is which).

so C+D = odd

KDM91 · Answer

I like the solution - it’s helpful.

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