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If, for all positive integer values of n, P(n) is defined as

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If, for all positive integer values of n, P(n) is defined as [#permalink] New post 25 Apr 2013, 21:43
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If, for all positive integer values of n, P(n) is defined as the sum of the smallest n prime numbers, then which of the following quantities are odd integers?

I. P(10)
II. P(P(10))
III. P(P(P(10)))

(A) I only
(B) I and II only
(C) I and III only
(D) II and III only
(E) I, II, and III
[Reveal] Spoiler: OA

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Re: If, for all positive integer values of n, P(n) is defined as [#permalink] New post 25 Apr 2013, 22:41
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emmak wrote:
If, for all positive integer values of n, P(n) is defined as the sum of the smallest n prime numbers, then which of the following quantities are odd integers?
I. P(10)
II. P(P(10))
III. P(P(P(10)))

(A) I only
(B) I and II only
(C) I and III only
(D) II and III only
(E) I, II, and III

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2 is the only even prime number. Now, the sum of first 10 prime numbers = 9*odd + 2 = odd. Thus P(10) = odd.

Again, the sum of the first n -odd primes = (n-1)*odd+2 = even.Thus, p(P(10)) = even.

Similarly, the sum of the first m - even primes = (m-1)*odd+2 = odd.

I and III.

C.
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Re: If, for all positive integer values of n, P(n) is defined as [#permalink] New post 25 Apr 2013, 23:03
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emmak wrote:
If, for all positive integer values of n, P(n) is defined as the sum of the smallest n prime numbers, then which of the following quantities are odd integers?
I. P(10)
II. P(P(10))
III. P(P(P(10)))

(A) I only
(B) I and II only
(C) I and III only
(D) II and III only
(E) I, II, and III

Express appreciation by pressing KUDOS button


This kind of question can give students fits. The key is to figure out what is being asked. The function P is summing up the first N prime numbers, so for example P(3) = 2 + 3 + 5. The total is 10. If I picked P(4), I'd get the same sum + 7, or 17.

Since the question hinges on whether the sum is even or odd, the only number that's unique in this circumstance is 2, as it is the only even prime number in an otherwise homogenous sea of odd numbers. Thus P(1) is even, P(2) is odd, P(3) is even again and P(4) is odd again. This is the pattern, so clearly P(10) will be 2 + nine odd numbers, so it will be odd. We can eliminate D.

P(P(10)) is where this starts getting interesting. You're doing the same test on a number we don't exactly know, but we know it must be odd. Since we know the pattern, the odd number will give us even. P(P(10)) will not be odd, eliminate B and E.

P(P(10))) will be the same function over a number we just calculated would be even. Hence it must be odd again. Eliminate A, the answer must be C.

Function questions are among the least understood questions on the GMAT, and this type of question can get people spending 3-4 minutes extrapolating numbers. If you understand the pattern using a small sample and reasoning, you can get this question right in under two minutes.

Hope this helps!
-Ron
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Re: If, for all positive integer values of n, P(n) is defined as [#permalink] New post 26 Apr 2013, 09:48
VeritasPrepRon wrote:
emmak wrote:
If, for all positive integer values of n, P(n) is defined as the sum of the smallest n prime numbers, then which of the following quantities are odd integers?
I. P(10)
II. P(P(10))
III. P(P(P(10)))

(A) I only
(B) I and II only
(C) I and III only
(D) II and III only
(E) I, II, and III

Express appreciation by pressing KUDOS button


This kind of question can give students fits. The key is to figure out what is being asked. The function P is summing up the first N prime numbers, so for example P(3) = 2 + 3 + 5. The total is 10. If I picked P(4), I'd get the same sum + 7, or 17.

Since the question hinges on whether the sum is even or odd, the only number that's unique in this circumstance is 2, as it is the only even prime number in an otherwise homogenous sea of odd numbers. Thus P(1) is even, P(2) is odd, P(3) is even again and P(4) is odd again. This is the pattern, so clearly P(10) will be 2 + nine odd numbers, so it will be odd. We can eliminate D.

P(P(10)) is where this starts getting interesting. You're doing the same test on a number we don't exactly know, but we know it must be odd. Since we know the pattern, the odd number will give us even. P(P(10)) will not be odd, eliminate B and E.

P(P(10))) will be the same function over a number we just calculated would be even. Hence it must be odd again. Eliminate A, the answer must be C.

Function questions are among the least understood questions on the GMAT, and this type of question can get people spending 3-4 minutes extrapolating numbers. If you understand the pattern using a small sample and reasoning, you can get this question right in under two minutes.

Hope this helps!
-Ron


Ah that makes sense... took me a while.

I. Even + 9 odds = odd.
II. Even + (odd * (odd - 1)) = even
III. Even + (odd * (even - 1)) = odd
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Re: If, for all positive integer values of n, P(n) is defined as [#permalink] New post 26 Feb 2014, 06:35
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Nice question +1. For this I would identify two cases: First we know that the only even number is 2. Therefore, if the number of prime integers that is n is even then the sum is odd, while if n is odd the sum is even.

In I we have that n is even therefore sum is odd. TRUE.

In II we have that the sum is odd so if we take that n is odd then the sum again will be even. FALSE.

In III, we get that the sum of n=10 is odd. Now the sum of odd is even and again the sum of even is odd. So III is TRUE as well.

Thus answer is C

Just to clarify

n=even, sum is odd
n=odd, sum is even

Hope this is clear
Cheers
J
Re: If, for all positive integer values of n, P(n) is defined as   [#permalink] 26 Feb 2014, 06:35
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