odds and evens - WOW.

[bblast]

If a,b,c ,d and 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

C

[Frankenstein]

Hi,
a-c+b is even. So, there are 2 possibilities
1)a,b,c are all even (or)
2)2 of them are odd, 1 is even

d+b-a is odd. So, there are 2 possibilities
3)a,b,d are all odd (or)
4)2 of them are even and 1 is odd

Combining the above cases, we get the following cases.
i)1 and 3 cannot be true at the same time.
ii)if 1 and 4 are true, we get a,b,c even and d is odd
iii)if 2 and 3 are true, we get a,b,d odd and c is even
iv)if 2 and 4 are true, we get c is odd, one of a,b is odd, d is even
Now, consider the options:
A) a+d is even in case (iii)
B) b+d is even in case (iii)
C) c+d is odd in all cases
D) a+b is even in case(ii) and case(iii)
E) a+c is even in case(ii) and case(iii)

Hence, C

[Ian Stewart]

If a,b,c ,d and 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

C

a - c + b is even, and d + b - a is odd, so if we subtract the first expression from the second expression, the result must be odd:

d + b - a - (a - c + b) = d + b - a - a + c - b = c + d - 2a

So c + d - 2a is odd. Since 2a is even, c + d must therefore be odd.

[bblast]

Ian's solution is as good as the question. Frank- I went by the method that you have nicely described in the post above, took me 3 min and 27 secs, :/

[Frankenstein]

Ian's solution is as good as the question. Frank- I went by the method that you have nicely described in the post above, took me 3 min and 27 secs, :/

Haha.. Good that you can save 3 mins now

[Tani]

I did this much more quickly. If a-c + b is even and d+b-a is odd then their sum is odd.

adding those two quantities you get d-c+2b. We know 2b is even, so d-c must be odd, which means d+c must be odd. 30 seconds!

[bblast]

I did this much more quickly. If a-c + b is even and d+b-a is odd then their sum is odd.

adding those two quantities you get d-c+2b. We know 2b is even, so d-c must be odd, which means d+c must be odd. 30 seconds!

Good 1 Tani !!
Yet, another subtle way to approach the problem.

Takeaway >> always try to manipulate given equations by addition/subtraction before diving into the answer choices for odds and evens.


i cant recall but there is a gmat prep question also which is on similar lines- rs+r = something

[amit2k9]

a-c+b-[b-a+d]= odd
thus
2a-(c+d)=odd
meaning c+d = odd