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# N is a positive integer, what is the probability that

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N is a positive integer, what is the probability that [#permalink]  20 Sep 2009, 14:14
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Let me lead this problem off with a question regarding number definitions:

Does the term "odd" imply automatically that the number is an integer? For example, 2.5 is not divisible by 2, so could it be called an odd number, just not an odd integer? Brain is fried and I'm not thinking clearly, so forgive me if this is a stupid question.

If n is a positive integer, what is the probability that a randomly selected factor of n will be even?

1.) n is even
2.) n/2 is odd

Answer Explanation is B... This is where my question comes into play.... If 9/2, for example, were considered to be an "odd" number, then you would either need statement 1 or you would need a specification that n/2 is an odd integer.

I am probably overanalyzing this though...
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Re: N is a positive integer, what is the probability that [#permalink]  20 Sep 2009, 14:32
Odd numbers are integers. I believe one of the properties of odd numbers is that, when divided by 2, it leaves a remainder of 1. But yeah, a lot of the books simply state that odd numbers are not divisible by 2.

As far as the question goes:

Statement 1: n is even:
Simply finding two number with contradictory probabilities (i.e. 2 and 4) which render this statement insufficient is fairly easy to do. Therefore options A and D are off the table.

Statement 2: n/2 is odd:
The first thing that strikes me is that, by claiming n/2 to be odd (and therefore an integer), n must be even. This is the same information provided in Statement 1, and therefore option C is off the table (this is one of the mental things I look for when doing DS questions. It helps me eliminate options).

Now let's consider the statement; we know that the number n is an even number, and that it has four definite factors: n (even), n/2 (odd), 2 (even), 1 (odd). Therefore we have 2 odd and 2 even factors of n. The next part is kind of tricky to explain without drawing it out, but I'll give it a shot anyways.

The number '2' cannot be broken down into further factors. It is possible however, that (n/2) has additional factors. It is provided, however, that n/2 is an odd number; as a result n/2 will only have odd factors (odd numbers can only be products of 2 odd numbers). Any combination of two numbers which have a product of n/2, however, will also create two additional even factors. For example, say X * Y = n/2. Therefore X and Y are two factors, both odd, of the number n. X*2 and Y*2, however, will also become 2 even factors of n, since n/X = Y*2, and n/Y = X*2.

Therefore, the probability is 50%, and the final answer is B.
Re: N is a positive integer, what is the probability that   [#permalink] 20 Sep 2009, 14:32
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# N is a positive integer, what is the probability that

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