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# If a and b are positive integers such that a-b and a/b are

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Manager
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If a and b are positive integers such that a-b and a/b are [#permalink]

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25 Nov 2008, 06:00
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Difficulty:

95% (hard)

Question Stats:

44% (02:27) correct 56% (01:48) wrong based on 223 sessions

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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

[Reveal] Spoiler:
I dont understand the answer choices here.
If I substitute 8 and 4 for a and b, i get both D&E as odd.
If I assume D&E to be even, both A and B need to be odd.
I just gave up and guessed C (not saying C is the OA)

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-a-and-b-are-positive-integers-such-that-a-b-and-a-b-are-88108.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 14 Sep 2014, 15:39, edited 1 time in total.
Renamed the topic, edited the question, added the OA and moved to PS forum.
Manager
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Re: If a and b are positive integers such that a-b and a/b are [#permalink]

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25 Nov 2008, 06:33
twilight wrote:
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

I dont understand the answer choices here.
If I substitute 8 and 4 for a and b, i get both D&E as odd.
If I assume D&E to be even, both A and B need to be odd.
I just gave up and guessed C (not saying C is the OA)

I do not have the answer yet but I would like to comment on your reasoning. I think D and E can be odd or even, even when a and b are even. If you take a = 6, then you will have (6+2)/2 = 4 even. Or you can take b = 10, then you will have (10+2)/2 = 6 even.
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Re: If a and b are positive integers such that a-b and a/b are [#permalink]

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25 Nov 2008, 06:36
Agree with you.
What I was poitning out there (with the 8 and 4 example) was that for two even numbers that satisfy the question, I get a situation where two of the choices are potentials answers.

---edit---
but I see the flaw in my reasoning here, since the question asks for a 'must be odd' response, so substitution is not the right way to go.
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Re: If a and b are positive integers such that a-b and a/b are [#permalink]

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25 Nov 2008, 06:48
good question, and it initially confussed me

to satisfy a-b and a/b are both even integers, a and b should be both even.

A - a can be 2 (a/2=1) or 4 (a/2=2) Out
B - the same as A Out
C - (4+2)/2 = 3 or (8+4)/2=6
E - The same as in A,B

Thus, only D satisfies, because a can't be equal to 2

Last edited by atletikos on 25 Nov 2008, 13:27, edited 1 time in total.
Manager
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Re: If a and b are positive integers such that a-b and a/b are [#permalink]

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25 Nov 2008, 07:29
I did not consider that 'a' must not be 2, which is true and your explantion makes sense.

However, theres nothing stopping a from being say a 6 - and in that case (a+2)/2 would be even, and hence would not agree with the requirement of 'must be odd'..

what say?
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Re: If a and b are positive integers such that a-b and a/b are [#permalink]

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25 Nov 2008, 08:18
I get D too...basically A >2 and could be of the form 2*N...so 2*n/2=n+1 is always odd.
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Re: If a and b are positive integers such that a-b and a/b are [#permalink]

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25 Nov 2008, 08:25
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twilight wrote:
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

I dont understand the answer choices here.
If I substitute 8 and 4 for a and b, i get both D&E as odd.
If I assume D&E to be even, both A and B need to be odd.
I just gave up and guessed C (not saying C is the OA)

7-t66732
I think this prob and yours are the same.
a/b is even so a must be even
a+b is even so b must be even
a/b is even while b is even so a must be a multiple of 4.
a is a multiple of 4 => a+2 is not a multiple of 4. => (a+2)/2 must be odd.
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Re: If a and b are positive integers such that a-b and a/b are [#permalink]

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25 Nov 2008, 08:38
fresinha12 wrote:
I get D too...basically A >2 and could be of the form 2*N...so 2*n/2=n+1 is always odd.

But if N is odd (which it can be), then 2*n/2=n+1 will be even?
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Re: If a and b are positive integers such that a-b and a/b are [#permalink]

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25 Nov 2008, 08:40
DavidArchuleta wrote:

7-t66732
I think this prob and yours are the same.
a/b is even so a must be even
a+b is even so b must be even
a/b is even while b is even so a must be a multiple of 4.
a is a multiple of 4 => a+2 is not a multiple of 4. => (a+2)/2 must be odd.

ah yes!
thats perfect, thanks.
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Re: If a and b are positive integers such that a-b and a/b are [#permalink]

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25 Nov 2008, 08:41
twilight wrote:
fresinha12 wrote:
I get D too...basically A >2 and could be of the form 2*N...so 2*n/2=n+1 is always odd.

But if N is odd (which it can be), then 2*n/2=n+1 will be even?

this is not possible sinc a/b is an even integer which means N is even..
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Re: If a and b are positive integers such that a-b and a/b are [#permalink]

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25 Nov 2008, 21:31
DavidArchuleta wrote:
twilight wrote:
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

I dont understand the answer choices here.
If I substitute 8 and 4 for a and b, i get both D&E as odd.
If I assume D&E to be even, both A and B need to be odd.
I just gave up and guessed C (not saying C is the OA)

7-t66732
I think this prob and yours are the same.
a/b is even so a must be even
a+b is even so b must be even
a/b is even while b is even so a must be a multiple of 4.
a is a multiple of 4 => a+2 is not a multiple of 4. => (a+2)/2 must be odd.

+1 David. Good explanation
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Re: If a and b are positive integers such that a-b and a/b are [#permalink]

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28 Nov 2008, 20:08
You may use substitution:

Initially substitute a and b with 8 and 2 respectively.
a = 8
b = 2

8 - 2 = even number
8/2 = even number

so these numbers satisfy requirements

a) a/2: 8/2 = 4 EVEN

b) b/2: 4/2 = 2 EVEN

c) (a+b)/2: (8+4)/2 = 6 EVEN

d) (a+2)/2: (8+2)/2 = 5 ODD
substitute another set (a=4 & b=2) for check
(4+2)/2 = 3 Again, ODD

e) (b+2)/2: (4+2)/2 = 3 ODD
substitute another set (a=4 & b=2) for check
(2+2)/2 = 2 EVEN

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Re: If a and b are positive integers such that a-b and a/b are [#permalink]

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14 Sep 2014, 08:12
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Re: If a and b are positive integers such that a-b and a/b are [#permalink]

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14 Sep 2014, 15:39
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twilight wrote:
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

[Reveal] Spoiler:
I dont understand the answer choices here.
If I substitute 8 and 4 for a and b, i get both D&E as odd.
If I assume D&E to be even, both A and B need to be odd.
I just gave up and guessed C (not saying C is the OA)

$$a-b$$ 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.

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-a-and-b-are-positive-integers-such-that-a-b-and-a-b-are-88108.html
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Re: If a and b are positive integers such that a-b and a/b are   [#permalink] 14 Sep 2014, 15:39
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