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# If a and b are positive integers

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Intern
Joined: 15 Sep 2017
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If a and b are positive integers [#permalink]  09 Sep 2018, 21:20
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Question Stats:

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.
Formatted
Intern
Joined: 27 Aug 2018
Posts: 36
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Kudos [?]: 16 [0], given: 7

Re: If a and b are positive integers [#permalink]  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
Intern
Joined: 15 Sep 2017
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Kudos [?]: 15 [0], given: 2

Re: If a and b are positive integers [#permalink]  10 Sep 2018, 08:00
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]  10 Sep 2018, 08:20
4
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Expert's post
$$a-b$$ 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.

Hope this helps. The answer is one and only.

Regards
_________________
Intern
Joined: 27 Aug 2018
Posts: 36
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Kudos [?]: 16 [0], given: 7

Re: If a and b are positive integers [#permalink]  10 Sep 2018, 08:40
AchyuthReddy 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
Intern
Joined: 19 Mar 2018
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Kudos [?]: 8 [0], given: 11

Re: If a and b are positive integers [#permalink]  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?
Intern
Joined: 27 Oct 2018
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Kudos [?]: 8 [1] , given: 17

Re: If a and b are positive integers [#permalink]  02 Nov 2018, 23:24
1
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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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