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What is the probability of selecting a clean number from a s

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What is the probability of selecting a clean number from a s [#permalink] New post 26 Sep 2012, 10:35
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What is the probability of selecting a clean number from a set of integers containing all multiples of 3 between 1 and 99, inclusive?

1. A clean number is an integer divisible by only 2 factors, one of which is greater than 2.
2. A clean number must be odd.

Source : Veritas

I am not sure why the OA is
[Reveal] Spoiler:
A
. Knowing that there are a finite number of odd integers between 1 and 99, B should be sufficient...
[Reveal] Spoiler: OA
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Re: What is the probability of selecting a clean number from a s [#permalink] New post 26 Sep 2012, 12:35
Expert's post
voodoochild wrote:
What is the probability of selecting a clean number from a set of integers containing all multiples of 3 between 1 and 99, inclusive?

1. A clean number is an integer divisible by only 2 factors, one of which is greater than 2.
2. A clean number must be odd.

Source : Veritas

I am not sure why the OA is
[Reveal] Spoiler:
A
. Knowing that there are a finite number of odd integers between 1 and 99, B should be sufficient...

I don't like this question. The prompt is clear --- at least the set from which we are choosing is perfectly clear. It's a least a clever idea in theory, introducing a brand new term, "clean numbers", in a DS questions, but I think the requirement of a definition and the structure of the DS conflict in ways the question's authors didn't anticipate.

A number with only two factors --- that's a prime number, numbers with a factor of only 1 and the number. By specifying one factor has to be greater than 2, we are specifying a prime number greater than 2. My question: is this sufficient for the definition of a clean number? is this necessary for the definition of a clean number?
Yes, I recognize the irony, Voodoo, my friend. Having recently persuaded you of the superfluity of these two words in CR, here I am using them in DS. To be fair, I am only using them in what I consider a poorly written DS --- a well-written DS wouldn't raise these questions.
Statement #1 says, essentially, clean number is a prime number greater than 3.
Is this a necessary statement --- only prime numbers greater than three are in the set of clean numbers?
Or, we equating clean numbers with the set of all prime numbers greater than 3? In other words, are we justified in assuming that statement #1 is a full definition of a clean number? That's somewhat unclear.

The second statement is even worse: "A clean number must be odd."
Is that saying ---- (necessary) ---- if I number is clean, it is odd?
Or, is it saying -----(sufficient) ---- if I number is odd, then its clean?

I gather, from your selection of (B) as the answer, that you interpreted the second statement as sufficient. The problem with that is: it's always the case that the two statements of a DS question have to be mathematically consistent. Since statement #1 restricts us to primes, it can't be every odd number --- that would be inconsistent. We have to take the "necessary" interpretation of statement #2.
Then, given the OA, it appears that we decide on either the "necessary" or "sufficient" interpretation of statement #1, then it would be sufficient to answer the question,and the answer would be (A). BUT, if that decision itself is something left to the reader, and no further information is given, then the answer would be (E).

My guess is that the authors of this question were not even thinking about "necessary" and "sufficient" --- they were way too naive in trying to write a "definition of a new term" question in DS form, and did not consider all the ramifications.

Voodoo, I assure you, on no legitimately constructed GMAT Quant question will you ever have even to think for a moment about the ideas of "necessary" and "sufficient" --- we only have to have recourse to them to discuss the inadequacies of questions such as this.

Does all this make sense?

Mike :-)
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Re: What is the probability of selecting a clean number from a s [#permalink] New post 27 Sep 2012, 17:57
mikemcgarry wrote:
The second statement is even worse: "A clean number must be odd."
Is that saying ---- (necessary) ---- if I number is clean, it is odd?
Or, is it saying -----(sufficient) ---- if I number is odd, then its clean?

I gather, from your selection of (B) as the answer, that you interpreted the second statement as sufficient. The problem with that is: it's always the case that the two statements of a DS question have to be mathematically consistent. Since statement #1 restricts us to primes, it can't be every odd number --- that would be inconsistent. We have to take the "necessary" interpretation of statement #2.

Mike,
Thanks for your help. I think that Veritas guys are testing CR skills in this question. (I have seen some official GMAT Quant questions that actually do that. I don't blame them). However, I am a bit confused. In my opinion, both the statements are necessary conditions. How? (one uses 'must be' and the other one states a fact about any general integer). None of the conditions are sufficient.

Secondly, let's assume, for our analysis, that the two conditions are sufficient. Still, a sufficient condition guarantees a specific outcome. It is not "THE sufficient condition." For instance, I could say that a 'troublesome' number is a square of 5, or a 'troublesome' number is any integer greater than 20 but less than 30 that has an integer square root. There could be other definitions or sufficient conditions for 'clean number.' Thus, there could multiple sufficient conditions.

I didn't follow your explanation about inconsistency. Can you please clear that for me?

Thanks in advance....
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Re: What is the probability of selecting a clean number from a s [#permalink] New post 28 Sep 2012, 03:23
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This question is deeply problematic, logically speaking, and you could never see a real GMAT question that resembles this one. The first reason I find it logically nonsensical is that it is not clear that we need a definition of 'clean number' at all to answer the question. If instead you were asked "What is the probability of selecting a prime number from a set of integers containing all multiples of 3 between 1 and 99, inclusive?" you obviously wouldn't need any additional information to solve. You wouldn't even need to know what a prime number was - you'd just need to know that prime numbers have some kind of definition, and that the question is therefore solvable. The same is true here. When I read this question, I think "well, I don't know what 'clean numbers' are, but as long as someone knows, the question can be answered in theory, so why do I need any statements at all?" After all, DS isn't testing if you can answer the question - it's testing if the question has only one answer. So that's the first problem: the question needs to make clear that 'clean numbers' are not something you could learn about if you read more math books, and are instead something the question has just invented on the spot.

That issue is bad enough, but then as Mike pointed out, it further isn't clear whether each statement is giving a sufficient condition or only a necessary one. The wording of Statement 1, which reads "A clean number is an integer divisible by only 2 factors, one of which is greater than 2", would be true if 17 was the only clean number, and would be true if all odd primes were clean numbers. It certainly is not sufficient to define the set of clean numbers, and if the OA is A, then the question writer was very confused on this point. Of course the same issue afflicts Statement 2.

So it doesn't make any sense to discuss what the answer to this question is, because the question makes no logical sense in the first place.
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Re: What is the probability of selecting a clean number from a s [#permalink] New post 01 Oct 2012, 12:40
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voodoochild wrote:
Mike,
Thanks for your help. In my opinion, both the statements are necessary conditions. How? (one uses 'must be' and the other one states a fact about any general integer). None of the conditions are sufficient.
Secondly, let's assume, for our analysis, that the two conditions are sufficient. Still, a sufficient condition guarantees a specific outcome. It is not "THE sufficient condition." For instance, I could say that a 'troublesome' number is a square of 5, or a 'troublesome' number is any integer greater than 20 but less than 30 that has an integer square root. There could be other definitions or sufficient conditions for 'clean number.' Thus, there could multiple sufficient conditions.
I didn't follow your explanation about inconsistency. Can you please clear that for me?
Thanks in advance....

Voodoo
First of all, I heartily applaud what IanStewart says. This is a poor question, and does not merit attention in and of itself. I am responding only to answer your questions.

Statement #1 equates "clean numbers" with the set of odd prime numbers --- again, the whole set, or only part of the set? We don't know.

Statement #2 the interpretation "if it's a clean number, then it's odd" --- that's actually redundant with statement #1: if clean numbers are the set of odd primes, or some subset thereof, then of course they have to be odd.
By contrast, the interpretation "if the number is odd, then it's a clean number" --- that's the inconsistency of which I spoke, because then "clean numbers" would include 9, 15, 21, 27, 33, 35, etc. all kinds of odd numbers that are not prime. This would contradict statement #1, which shouldn't happen in a DS question. (Although, in a question of this poor quality, I suppose all bets are off.)

Does that make sense?

Mike :-)
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Re: What is the probability of selecting a clean number from a s [#permalink] New post 01 Oct 2012, 14:33
Ian and Mike - thanks for your comments. I have deleted this example from my memory. In fact, Veritas has two such examples in their book! :)
Re: What is the probability of selecting a clean number from a s   [#permalink] 01 Oct 2012, 14:33
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