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If a and b are positive integers

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If a and b are positive integers [#permalink] New post 09 Sep 2018, 21:20
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15% (01:08) correct 84% (01:29) wrong based on 26 sessions
If a and b are positive integers such that \(a-b\) and \(\frac{a}{b}\) are both even integers. Which of the following must be an odd integer?

A) \(\frac{a}{2}\)

B) \(\frac{b}{2}\)

C) \(\frac{a+b}{2}\)

D) \(\frac{a+2}{2}\)

E) \(\frac{b+2}{2}\)
[Reveal] Spoiler: OA

Last edited by Carcass on 10 Sep 2018, 03:49, edited 1 time in total.
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Re: If a and b are positive integers [#permalink] New post 10 Sep 2018, 06:23
The answers are D and E.

Since a-b = even and a/b = even, then both a and b are positive even integers.

We can rewrite D as a/2 + 2/2. a/2 will always be even, 2/2 = 1. Even+1 = Odd.

The same for E
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Re: If a and b are positive integers [#permalink] New post 10 Sep 2018, 08:00
Nomad wrote:
The answers are D and E.

Since a-b = even and a/b = even, then both a and b are positive even integers.

We can rewrite D as a/2 + 2/2. a/2 will always be even, 2/2 = 1. Even+1 = Odd.

The same for E


Yeah, I too got same doubt,Just posted the question with same answer as in material.
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Re: If a and b are positive integers [#permalink] New post 10 Sep 2018, 08:20
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\(a-b\) even --> either both even or both odd

\(\frac{a}{b}\) even --> either both even or \(a\) is even and \(b\) is odd.

As both statements are true --> \(a\) and \(b\) must be even.

As \(\frac{a}{b}\) is an even integer --> \(a\) must be multiple of 4.


Options A is always even.

Options B can be even or odd.

Options C can be even or odd.

Options D: \(\frac{a+2}{2}=\frac{a}{2}+1\), as \(a\) is multiple of \(4\), \(\frac{a}{2}\) is even integer --> even+1=odd. Hence option D is always odd.

Options E can be even, odd.


Answer: D.

Hope this helps. The answer is one and only.

Regards
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Intern
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Re: If a and b are positive integers [#permalink] New post 10 Sep 2018, 08:40
AchyuthReddy wrote:
Nomad wrote:
The answers are D and E.

Since a-b = even and a/b = even, then both a and b are positive even integers.

We can rewrite D as a/2 + 2/2. a/2 will always be even, 2/2 = 1. Even+1 = Odd.

The same for E


Yeah, I too got same doubt,Just posted the question with same answer as in material.


Actually, @Carcass solution is the correct one. I did not consider if b=2. That is a mistake
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Re: If a and b are positive integers [#permalink] New post 13 Sep 2018, 12:08
How is E wrong?
if a = 4, b = 2

then b--> 2+2/2 = 2 which is even?
Please explain what is wrong here?
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Re: If a and b are positive integers [#permalink] New post 02 Nov 2018, 23:24
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a must be greater than 2 in a way a/b is even.
b can be 2 also in which case b+2/2 is NOT odd
Re: If a and b are positive integers   [#permalink] 02 Nov 2018, 23:24
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