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TAGS: Intern Joined: 15 Sep 2017
Posts: 34
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Kudos [?]: 17  , given: 2

If a and b are positive integers [#permalink]
3
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Question Stats: 34% (01:01) correct 65% (01:38) wrong based on 70 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
Followers: 0

Kudos [?]: 21 , given: 7

Re: If a and b are positive integers [#permalink]
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
Posts: 34
Followers: 0

Kudos [?]: 17 , given: 2

Re: If a and b are positive integers [#permalink]
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. Founder  Joined: 18 Apr 2015
Posts: 12594
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Kudos [?]: 3164  , given: 11624

Re: If a and b are positive integers [#permalink]
6
KUDOS
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
_________________

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Intern Joined: 27 Aug 2018
Posts: 36
Followers: 0

Kudos [?]: 21 , given: 7

Re: If a and b are positive integers [#permalink]
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
Manager Joined: 19 Mar 2018
Posts: 58
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Kudos [?]: 23 , given: 16

Re: If a and b are positive integers [#permalink]
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
Posts: 49
Followers: 0

Kudos [?]: 19  , given: 27

Re: If a and b are positive integers [#permalink]
1
KUDOS
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
Intern Joined: 11 Aug 2019
Posts: 2
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Kudos [?]: 1 , given: 0

Re: If a and b are positive integers [#permalink]
Carcass wrote:
$$a-b$$ 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

what if a=12 and b=4?
then a/b will not be even. plz explain. thanks in advance Re: If a and b are positive integers   [#permalink] 04 Oct 2019, 19:27
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