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

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Senior 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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18 Sep 2007, 12:57
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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

Man I hate that the copy pasting drops all formatting!!! I have fixed the question now. Sorry for wasting all your time.

Last edited by gluon on 18 Sep 2007, 17:19, edited 2 times in total.

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Director
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Re: PS: Set 7, Q36 - Odd and even numbers [#permalink]

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18 Sep 2007, 13:25
gluon wrote:
If a and b are positive integers such that a – b and ba 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

No clue.... Is this question correct?

- Brajesh

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Director
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Re: PS: Set 7, Q36 - Odd and even numbers [#permalink]

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18 Sep 2007, 13:33
gluon wrote:
If a and b are positive integers such that a – b and ba 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

Can`t find the right answer choice among given either,

a-b=even
ab=even

means there are four possible scenarios for a and b:

a=even;b=even
a=zero;b=even
a=even;b=zero
a=zero;b=zero

if u work through the answer choices non of them must necessarily be odd.

unless i am missing something...

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Director
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Re: PS: Set 7, Q36 - Odd and even numbers [#permalink]

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18 Sep 2007, 13:37
gluon wrote:
If a and b are positive integers such that a – b and ba 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

This question has wrong choices. I remember this question from some test . These have wrong options for sure.

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CEO
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Re: PS: Set 7, Q36 - Odd and even numbers [#permalink]

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18 Sep 2007, 21:08
gluon 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

Man I hate that the copy pasting drops all formatting!!! I have fixed the question now. Sorry for wasting all your time.

Something isn't right w/ this question. Ignore a/b b/c there are no definite rules about odds and evens for division.

but if a-b is even then a and b are both even or both odd.

A: a can be even so a/2 can be even, not must be odd.
B. same reason as A
C. since a and b can both be even then this doesnt have to be odd
D. a+2/2 a can be even so this doesnt have to be odd.
E. Same reason as D.

?_?

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CEO
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Re: PS: Set 7, Q36 - Odd and even numbers [#permalink]

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18 Sep 2007, 21:09
IrinaOK wrote:
gluon wrote:
If a and b are positive integers such that a – b and ba 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

Can`t find the right answer choice among given either,

a-b=even
ab=even

means there are four possible scenarios for a and b:

a=even;b=even
a=zero;b=even
a=even;b=zero
a=zero;b=zero

if u work through the answer choices non of them must necessarily be odd.

unless i am missing something...

careful with your wording here. just an FYI. a and b are BOTH positive integers so 0 is out.

I always do this >:-X

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Director
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19 Sep 2007, 04:11
I would go for 'D'.

a and b both should be even to satusfy the a-b and a/b to be even.
As a/b is even, a>b. The minimum value B can acquire is 2. So a will be greater than 2.

Now close call is between D and E.
a/2 +1 will always be odd as A/2 is even.

b/2 + 1 not neccessarily be odd. For b =2, it's not odd.

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Senior Manager
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19 Sep 2007, 15:27

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Manager
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20 Sep 2007, 02:54

(a+2)/2 has to be odd.

a/b is even - means a is even
a-b is even - means since a is even b is also even.

now, b is even and a/b is even meaning a should be a multiple of 4.

if a is a multiple of 4 then a+2 will not be a multiple of 4. hence (a+2)/2 is odd.

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Manager
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Re: PS: Set 7, Q36 - Odd and even numbers [#permalink]

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20 Sep 2007, 04:25
gluon 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

Man I hate that the copy pasting drops all formatting!!! I have fixed the question now. Sorry for wasting all your time.

I choose D as well. if a=80 and b=40 then a/2 and b/2 are even. Therefore, A and B are incorrect. If a/2 and b/2 are even, then (a+b)/2 = a/2 + b/2 will be even as well. C is incorrect as well.

I am puzzled because it seems to me that (a+2)/2 = a/2 + 1 will be always odd. The same holds for (b+2)/2. But how can we say that D and E are correct? Am I missing something here?

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Manager
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20 Sep 2007, 04:41
vshaunak@gmail.com wrote:
I would go for 'D'.

a and b both should be even to satusfy the a-b and a/b to be even.
As a/b is even, a>b. The minimum value B can acquire is 2. So a will be greater than 2.

Now close call is between D and E.
a/2 +1 will always be odd as A/2 is even.

b/2 + 1 not neccessarily be odd. For b =2, it's not odd.

vshaunak if b=2 then (a+b)/2 = a/2 + 1, which will be always odd and therefore C gets correct...

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Director
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Re: PS: Set 7, Q36 - Odd and even numbers [#permalink]

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20 Sep 2007, 09:01
gluon 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

Man I hate that the copy pasting drops all formatting!!! I have fixed the question now. Sorry for wasting all your time.

The stem tells us that a-b is even which means that a and b are either both odd or both even.

The stem also tells us that a/b is even (a/b=E). If you multiply b by both sides you see that a =b*E. In this case a has to be even because the product of any number and an even number is even.

To satisfy both requirements a and b must be even.

A. Even/2 could be odd or even (Ex. 8/2 = 4, 14/2=7)
B. same as A
C. same as A
D. This has to be odd becasue a has to be a multiple of 4 as posted earlier.
E. This could both be even or odd as well. Also note b/2 + 1 does NOT have to be odd because a/2 can be odd or even therefore and even + odd = odd and odd + odd = even.

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Director
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20 Sep 2007, 10:01
robinantony wrote:

(a+2)/2 has to be odd.

a/b is even - means a is even
a-b is even - means since a is even b is also even.

now, b is even and a/b is even meaning a should be a multiple of 4.

if a is a multiple of 4 then a+2 will not be a multiple of 4. hence (a+2)/2 is odd.

Great explanation dude. Now D makes sense over C. I guess the trick is to figure out that 'a' has to be a multiple of 4.

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22 Sep 2007, 05:48
after15 wrote:
vshaunak@gmail.com wrote:
I would go for 'D'.

a and b both should be even to satusfy the a-b and a/b to be even.
As a/b is even, a>b. The minimum value B can acquire is 2. So a will be greater than 2.

Now close call is between D and E.
a/2 +1 will always be odd as A/2 is even.

b/2 + 1 not neccessarily be odd. For b =2, it's not odd.

vshaunak if b=2 then (a+b)/2 = a/2 + 1, which will be always odd and therefore C gets correct...

We have to find the solution, which is always true. B can be 2 or some other even integer. If it's working for b=2 does not necessarily mean that it will work for other values.

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22 Sep 2007, 05:48
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