BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

Product of primes

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
Master | Next Rank: 500 Posts
Posts: 113
Joined: Wed Dec 07, 2011 7:20 am
Thanked: 3 times

Product of primes

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!
Top
Source: — Problem Solving |
User avatar
Legendary Member
Posts: 2663
Joined: Wed Jan 14, 2015 8:25 am
Location: Boston, MA
Thanked: 1153 times
Followed by:128 members
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.
Veritas Prep | GMAT Instructor

Veritas Prep Reviews
Save $100 off any live Veritas Prep GMAT Course
Top

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 16207
Joined: Mon Dec 08, 2008 6:26 pm
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1268 members
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
Image
Top
User avatar
GMAT Instructor
Posts: 15539
Joined: Tue May 25, 2010 12:04 pm
Location: New York, NY
Thanked: 13060 times
Followed by:1906 members
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.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Top
Post new topic Post Reply
• Page 1 of 1