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If a and b are positive integers such that ab and a/b are [#permalink]
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25 Nov 2008, 06:00
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If a and b are positive integers such that ab 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) OPEN DISCUSSION OF THIS QUESTION IS HERE: ifaandbarepositiveintegerssuchthatabandabare88108.html
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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.



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Re: If a and b are positive integers such that ab 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 ab 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 ab 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 ab and a/b are [#permalink]
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25 Nov 2008, 06:48
good question, and it initially confussed me to satisfy ab 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.



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Re: If a and b are positive integers such that ab and a/b are [#permalink]
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25 Nov 2008, 07:29
Interesting answer, thanks for that. 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 ab 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 ab 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 ab 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) 7t66732I 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 ab 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 ab and a/b are [#permalink]
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25 Nov 2008, 08:40
DavidArchuleta wrote: 7t66732I 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 ab 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 ab 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 ab 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) 7t66732I 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 ab 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
ANSWER: D



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Re: If a and b are positive integers such that ab 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 ab 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) \(ab\) 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. Answer: D. OPEN DISCUSSION OF THIS QUESTION IS HERE: ifaandbarepositiveintegerssuchthatabandabare88108.html
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