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bblast: Legendary Member; Posts: 1077; Joined: Mon Dec 13, 2010 1:44 amFollowed by: 33 members; Thanked: 118 times; GMAT Score:710

by bblast » Sun Jul 10, 2011 7:10 am

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

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Frankenstein: Legendary Member; Posts: 1448; Joined: Tue May 17, 2011 9:55 amFollowed by: 53 members; Location: India; Thanked: 375 times

by Frankenstein » Sun Jul 10, 2011 9:05 am

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

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

by Ian Stewart » Sun Jul 10, 2011 9:32 am

bblast wrote: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.

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bblast: Legendary Member; Posts: 1077; Joined: Mon Dec 13, 2010 1:44 amFollowed by: 33 members; Thanked: 118 times; GMAT Score:710

by bblast » Sun Jul 10, 2011 9:54 am

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, :/

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Frankenstein: Legendary Member; Posts: 1448; Joined: Tue May 17, 2011 9:55 amFollowed by: 53 members; Location: India; Thanked: 375 times

by Frankenstein » Sun Jul 10, 2011 10:08 am

bblast wrote: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

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Tani: Legendary Member; Posts: 1255; Joined: Fri Nov 07, 2008 2:08 pmFollowed by: 90 members; Location: St. Louis; Thanked: 312 times

by Tani » Sun Jul 10, 2011 10:39 am

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!

Tani Wolff

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bblast: Legendary Member; Posts: 1077; Joined: Mon Dec 13, 2010 1:44 amFollowed by: 33 members; Thanked: 118 times; GMAT Score:710

by bblast » Sun Jul 10, 2011 8:31 pm

Tani Wolff - Kaplan wrote: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

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amit2k9: Master | Next Rank: 500 Posts; Posts: 461; Joined: Tue May 10, 2011 9:09 amFollowed by: 3 members; Location: pune; Thanked: 36 times

by amit2k9 » Mon Jul 11, 2011 9:24 pm

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

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