odd integer

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gibran: Senior | Next Rank: 100 Posts; Posts: 42; Joined: Sat Apr 12, 2008 8:51 am; Thanked: 1 times

by gibran » Thu May 22, 2008 9:05 am

If a and b are positive integers such that a – b and a/b are both even integers, which of the following must be an odd integer?

A: a/2
B: b/2
C: (a+b)/2
D: (a+2)/2
E: (b+2)/2

OA is D. Please explain with steps.

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Source: Beat The GMAT — Data Sufficiency | Thu May 22, 2008 9:05 am

aatech: Master | Next Rank: 500 Posts; Posts: 192; Joined: Tue Apr 29, 2008 7:41 am; Thanked: 14 times

by aatech » Thu May 22, 2008 9:40 am

For a-b to be even, either both are even or both are odd

For a/b to be even, a has to be even

To satisfy both conditions, a and b both are even

Since, both are even none of the condtion given in question can be odd...

Anyone else????

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simplyjat: Master | Next Rank: 500 Posts; Posts: 423; Joined: Thu Dec 27, 2007 1:29 amFollowed by: 2 members; Location: Hyderabad, India; Thanked: 36 times; GMAT Score:770

by simplyjat » Thu May 22, 2008 9:49 am

We have to go a bit further. Even integer is an integer that is divisible by 2. A multiple of 4 is also an even integer but with the exception that the number when divided by 2 is again even number.

a - b is even means that either both a and b are even or both are odd.
a/b is even not only means that both are even, but also means that a has to be a multiple of 4. One can verify with some numbers.

The key here is understand the last point that a is multiple of 4.
(a + 2)/2 = a/2 + 1.
And as we know that a/2 is even a/2 + 1 must be odd...

simplyjat

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aatech: Master | Next Rank: 500 Posts; Posts: 192; Joined: Tue Apr 29, 2008 7:41 am; Thanked: 14 times

by aatech » Thu May 22, 2008 10:20 am

Question is asking for a condition that is always true (MUST).

a/2+1 will be odd only when a is multiple of 4

If you do the same logic for b, for that matter, E can be true also.

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egybs: Master | Next Rank: 500 Posts; Posts: 178; Joined: Wed May 14, 2008 3:51 pmFollowed by: 1 member; Thanked: 16 times; GMAT Score:99%+

by egybs » Thu May 22, 2008 12:07 pm

B has to be an even factor of A... in other words, A has to be a multiple of both 2 and another even number (or simply, divisible by 4). This isn't the case for B... it can be any even number.

If A is divisible by 4 and has B as a factor, A/2 +1 would always be odd.

Again, since B doesn't have to be divisible by 4, (B+2)/2 won't always be even.

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aatech: Master | Next Rank: 500 Posts; Posts: 192; Joined: Tue Apr 29, 2008 7:41 am; Thanked: 14 times

by aatech » Thu May 22, 2008 12:47 pm

Got it...

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