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m and n are positive integers. If m/n and m+n both are even,

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m and n are positive integers. If m/n and m+n both are even, [#permalink] New post 05 Jul 2008, 23:32
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m and n are positive integers. If m/n and m+n both are even, which of the following must be odd?

I. (m+n)/2
II. (m+2)/2
III. (n+2)/2

A. I only
B. II only
C. III only
D. I and II only
E. II and III only
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Re: Odd / Even question [#permalink] New post 05 Jul 2008, 23:50
It should be E (II and III only).

You can pick an example. Let m=16 and n=4 which satisfies the given condition of (m+n) and m/n being even.

I. (m+n)/2 = (20/2) = 10 (Even)- Hence not True
II. (m+2)/2 = (18/2) = 9 (odd)- Hence True
III. (n+2)/2 = (6/2) = 3 (odd) - Hence True

You can try this by another example (m=64, n=8), it should be true for (m,n)= (square of any even number, even number).
what is OA?
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Re: Odd / Even question [#permalink] New post 05 Jul 2008, 23:51
nirimblf wrote:
m and n are positive integers. If m/n and m+n both are even, which of the following must be odd?

I. (m+n)/2
II. (m+2)/2
III. (n+2)/2

A. I only
B. II only
C. III only
D. I and II only
E. II and III only


m/n is even so m must be even
m+n is even so n is even too
m/n is even while n is even so m is divisible by 4 so m/2 is always even

I. m/2+n/2 (nothing about n)
II. m/2 + 1 (always odd)
III. n/2+1 (nothing can be determined about n)

My choice is B.
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Re: Odd / Even question [#permalink] New post 05 Jul 2008, 23:59
m/n is even and m+n is even

so both m and n are even....

Also m/n = k (an even number) --> m = k*n (multiplication of two even numbers) --> m/2 will always be even
n/2 could be either even or odd

1. (m+n)/2 = m/2 + n/2 : either even or odd
2. (m+2)/2 = m/2 + 1 = e+1 : odd
3. (n+2)/2 = n/2 +1 " either even or Odd

Answer B
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Re: Odd / Even question [#permalink] New post 06 Jul 2008, 00:00
DavidArchuleta wrote:
m/n is even so m must be even
m+n is even so n is even too
m/n is even while n is even so m is divisible by 4 so m/2 is always even

I. m/2+n/2 (nothing about n)
II. m/2 + 1 (always odd)
III. n/2+1 (nothing can be determined about n)

My choice is B.


I don't understand. If n is even why nothing can be determined about n?
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Re: Odd / Even question [#permalink] New post 06 Jul 2008, 00:07
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nirimblf wrote:
DavidArchuleta wrote:
m/n is even so m must be even
m+n is even so n is even too
m/n is even while n is even so m is divisible by 4 so m/2 is always even

I. m/2+n/2 (nothing about n)
II. m/2 + 1 (always odd)
III. n/2+1 (nothing can be determined about n)

My choice is B.


I don't understand. If n is even why nothing can be determined about n?

I'm sorry, nothing can be determined about n/2. I'm careless as usual. Sorry again. :(
m/2 is always even 'coz m is divisible by 4 but n is just even, it can be divisible by 4 or not so n/2 is either even or odd.
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Re: Odd / Even question [#permalink] New post 06 Jul 2008, 00:34
I agree. I did not consider the possibility of n/2 being odd or even :( It should be B.
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Re: Odd / Even question [#permalink] New post 06 Jul 2008, 04:31
I don't understand why option B should be our answer. If M is even, then it doesn't necessarily mean that m/2 will be even. For example, 10/2 is 5, which is odd. so an even number divided by 2 could yield either even or odd. please explain guys! thanks
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Re: Odd / Even question [#permalink] New post 06 Jul 2008, 05:01
tarek99 wrote:
I don't understand why option B should be our answer. If M is even, then it doesn't necessarily mean that m/2 will be even. For example, 10/2 is 5, which is odd. so an even number divided by 2 could yield either even or odd. please explain guys! thanks

As it was said above, m is not only even, it is also divisible by 4.
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Re: Odd / Even question [#permalink] New post 06 Jul 2008, 14:49
nirimblf wrote:
m and n are positive integers. If m/n and m+n both are even, which of the following must be odd?

I. (m+n)/2
II. (m+2)/2
III. (n+2)/2

A. I only
B. II only
C. III only
D. I and II only
E. II and III only



m/n = Even
=> even/even , even/odd
m+n both are even
=>even+even, odd+odd

=>m,n both are even
m = even * n = even *even
=>m/2 = even

n/2 -not sure.

I. (m+n)/2 = even + not sure
II. (m+2)/2 = m/2 + 1 = even +odd = odd
III. (n+2)/2 = not sure +1

B
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Re: Odd / Even question [#permalink] New post 06 Jul 2008, 15:46
Oski wrote:
tarek99 wrote:
I don't understand why option B should be our answer. If M is even, then it doesn't necessarily mean that m/2 will be even. For example, 10/2 is 5, which is odd. so an even number divided by 2 could yield either even or odd. please explain guys! thanks

As it was said above, m is not only even, it is also divisible by 4.



i'm sorry if i might be asking a stupid question, but how do you know that m is divisible specifically by 4?? all we know is that m is even. So in statement II:

(m+2)/2 = (m/2) + (2/2) = (m/2) + 1

If m/2 is even, then + 1 will be odd

if m/2 is odd, then + 1 will be even. so where from the divisibility by 4??
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Re: Odd / Even question [#permalink] New post 06 Jul 2008, 23:52
tarek99 wrote:
i'm sorry if i might be asking a stupid question, but how do you know that m is divisible specifically by 4?? all we know is that m is even. So in statement II:

(m+2)/2 = (m/2) + (2/2) = (m/2) + 1

If m/2 is even, then + 1 will be odd

if m/2 is odd, then + 1 will be even. so where from the divisibility by 4??

It was said above:

m/n is even => m can be written as m = 2 * K * n, with K an integer i.e. m is even

Then m+n is even: since m is even then it tells us that n is even i.e. n can be written as n = 2 * L, with L an integer

Back to m, we can then write m = 2 * K * (2 * L) i.e. m = 4 * K * L => m is divisible by 4
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Re: Odd / Even question [#permalink] New post 07 Jul 2008, 05:16
tarek99 wrote:
Oski wrote:
tarek99 wrote:
I don't understand why option B should be our answer. If M is even, then it doesn't necessarily mean that m/2 will be even. For example, 10/2 is 5, which is odd. so an even number divided by 2 could yield either even or odd. please explain guys! thanks

As it was said above, m is not only even, it is also divisible by 4.



i'm sorry if i might be asking a stupid question, but how do you know that m is divisible specifically by 4?? all we know is that m is even. So in statement II:

(m+2)/2 = (m/2) + (2/2) = (m/2) + 1

If m/2 is even, then + 1 will be odd

if m/2 is odd, then + 1 will be even. so where from the divisibility by 4??


m /n = even
we know n is even
m = even * even
lest even number we know is 2
m = 2*2
we know m is atleast divisible by 4
Hope thi s helps
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Re: Odd / Even question [#permalink] New post 07 Jul 2008, 07:14
thanks a lot guys. that really helped! :)
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Re: Odd / Even question [#permalink] New post 07 Jul 2008, 08:04
durgesh79 wrote:
m/n is even and m+n is even

so both m and n are even....

Also m/n = k (an even number) --> m = k*n (multiplication of two even numbers) --> m/2 will always be even
n/2 could be either even or odd

1. (m+n)/2 = m/2 + n/2 : either even or odd
2. (m+2)/2 = m/2 + 1 = e+1 : odd
3. (n+2)/2 = n/2 +1 " either even or Odd

Answer B


THX for the awesome explanation.
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Re: Odd / Even question [#permalink] New post 12 Jul 2008, 12:18
great question..

agree with B...

took my about 2+ min to realize m has to be at least divisible by 4...

so 8+4=12/2=6 which is even
assuming n=4..

only 8+2/2 =even..
Re: Odd / Even question   [#permalink] 12 Jul 2008, 12:18
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