Product of primes

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manik11: Master | Next Rank: 500 Posts; Posts: 113; Joined: Wed Dec 07, 2011 7:20 am; Thanked: 3 times

by manik11 » Thu Dec 03, 2015 10:57 am

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
E) i and ii only
D) i and iii only
E) ii and iii only

OA : D
Source : Veritas Prep

I can figure out that X will always be odd because 2 will be there in any product of first N primes making the product Even and Even + Odd = Odd. This eliminates A and B. However I just can't figure out how to handle ii and iii. If I test values I get a prime value of X (at least in the limited number of test cases I've used) but clearly ii is not always true.
Experts...would really appreciate your help on this one.

Thanks!

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Source: Beat The GMAT — Problem Solving | Thu Dec 03, 2015 10:57 am

DavidG@VeritasPrep: Legendary Member; Posts: 2663; Joined: Wed Jan 14, 2015 8:25 amFollowed by: 128 members; Location: Boston, MA; Thanked: 1153 times; GMAT Score:770

by DavidG@VeritasPrep » Thu Dec 03, 2015 11:20 am

manik11 wrote: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

OA : D
Source : Veritas Prep

I can figure out that X will always be odd because 2 will be there in any product of first N primes making the product Even and Even + Odd = Odd. This eliminates A and B. However I just can't figure out how to handle ii and iii. If I test values I get a prime value of X (at least in the limited number of test cases I've used) but clearly ii is not always true.
Experts...would really appreciate your help on this one.

Thanks!

Instead of picking numbers to illustrate the statements, use a bit of logic.

You saw that (i) was true. So we can eliminate everything but C and D.

Now look at C. If C were correct, it would be the case that (ii) is true, and x is always prime, but that (iii) is not always true, and x COULD be the square of an integer. But that's impossible, as no prime could be the same square of an integer. So if (ii) is true, and x is prime, then (iii) must be true as well, as no prime is the square of an integer. Therefore it cannot be the case that (ii) is true and (iii) is not always true. So C is out. That leaves us with D.

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Thu Dec 03, 2015 11:28 am

Here's another practice question that features this interesting function - https://www.beatthegmat.com/p-12-11-t279341.html

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Thu Dec 03, 2015 1:13 pm

manik11 wrote: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

(i): x is always odd
Since the product of the first n prime numbers includes a factor of 2, p(n) = EVEN.
Thus, x = p(n) + 1 = EVEN + ODD = ODD.
Since (i) must be true, eliminate A, B and E.

(ii): x is always prime
x can be an INFINITELY LARGE integer.
While there are relatively easy ways to prove that a very large integer is NOT prime -- if the integer is even, if the integer is a multiple of 3, and so on -- there is no straightforward way for a test-taker to prove that a very large integer IS prime.
Thus, if the GMAT asks whether a very large integer is prime, the answer must be NO (since there is no straightforward way to provide that a very large integer IS prime).
Implication:
(ii) does not have to be true.
Eliminate C.

The correct answer is D.

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