Let the function p(n) represent the product of the first n p

Question

Let the function p(n) represent the product of the first n prime numbers, where n > 0. If x = p(n) + 1, which of the following must be true?

(i) x is always odd

(ii) x is always prime

(iii) x is never the square of an integer

A. ii only
B. iii only
C. i and ii only
D. i and iii only
E. ii and iii only

Bunuel · Accepted Answer

p(n) is always even, because the first prime is 2 and no matter what n is, 2 always will be a divisor of p(n). Thus, p(n) + 1 = even + 1 = odd. So, (i) is always true.

Now, use logic:

If (ii) is true (so if x is always prime), then (iii) must automatically be true: no prime is the square of an integer. So, the correct answer must be i only; i, ii, and iii only; or i and iii only. since only "i and iii only" is among the options, then it must be true.

Or, since (i) is always true, then from options the answer must be either C or D. C cannot be correct because if (ii) is true, then so must be (iii). Thus only D remains.

Answer: D.

dransa · Answer

A nice approach. 
I used the same approach. It took me 1 min 51 sec. I wonder if I could have figured the approach sooner though.

Gmatdecoder · Answer

I did not get how to choose between ii and iii. We know x to be prime. How can we dismiss ii then?

bhatiavai · Answer

You use the answer choices to make your decision. If ii is true than iii has to be true but we don't have any answer choice with all 3 options. Hence we go for i & iii

yenh · Answer

Hi Bunuel,
I'm just wondering whether ii is not true? Is there any case that makes ii is not true?

Thanks indeed

Bunuel · Answer

p(6) + 1 = 2*3*5*7*11*13+1=30031 = 59*509  is not a prime.

yenh · Answer

Thanks for your quick response. Btw, is there any way to be sure (theoretically) in case we cannot figure out by example?

Bunuel · Answer

There is no known formula for prime numbers (in fact it's one of the biggest math challenges), so p(n) + 1 cannot be prime for all values of n, else we would have the formula which gives primes.

yenh · Answer

Got it, thanks Bunuel.

Joc456 · Answer

try plugging in numbers. If n=2 then 2X3+1=7
if n=3 then 2*3*5=30+1=31 the answer will always be odd b/c 2 is a prime and the product will always be an even plus 1

If you try n=4 then you get 2*3*5*7=210 which is not a prime

D

GMATGuruNY · Answer

When the GMAT asks whether a value must be prime -- and that value can be extremely large -- the answer must be NO.
The reason:
The GMAT cannot expect us to prove that an extremely large number is prime.
Here, x can be INFINITELY LARGE.
Since the GMAT cannot expect us to prove that infinitely large options for x must be prime, we can conclude that Statement II does NOT have to be true.
Eliminate A, C and E.

Since the product of the first n prime numbers must include a factor of 2, p(n) = EVEN, with the result that p(n) + 1 = ODD.
Thus, Statement I must be true.
Eliminate B.

D

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Let the function p(n) represent the product of the first n p

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