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m and n are positive integers. If m/n and m+n both are even, [#permalink]
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06 Jul 2008, 00:32
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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]
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06 Jul 2008, 00:50
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]
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06 Jul 2008, 00:51
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]
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06 Jul 2008, 00:59
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]
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06 Jul 2008, 01:00
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]
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06 Jul 2008, 01:07
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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]
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06 Jul 2008, 01:34
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]
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06 Jul 2008, 05:31
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]
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06 Jul 2008, 06:01
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]
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06 Jul 2008, 15:49
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]
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06 Jul 2008, 16:46
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]
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07 Jul 2008, 00:52
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]
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07 Jul 2008, 06:16
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]
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07 Jul 2008, 08:14
thanks a lot guys. that really helped!



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Re: Odd / Even question [#permalink]
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07 Jul 2008, 09:04
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]
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12 Jul 2008, 13:18
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..




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