Question 2

This topic has expert replies
Master | Next Rank: 500 Posts
Posts: 164
Joined: Tue May 12, 2015 9:06 am
Thanked: 3 times

Question 2

by oquiella » Fri Sep 18, 2015 4:48 am
2. What is the greatest possible length of a positive integer less than 1,000?

Note: For any positive integer n, n>1, the "length" of n is the number of positive primes whose product is n. For example, the length of 50 is 3 since 50= (2) (5) (5)


A. 10

B. 9

C. 8

D. 7

E. 6

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 » Fri Sep 18, 2015 5:11 am
oquiella wrote:2. What is the greatest possible length of a positive integer less than 1,000?

Note: For any positive integer n, n>1, the "length" of n is the number of positive primes whose product is n. For example, the length of 50 is 3 since 50= (2) (5) (5)


A. 10

B. 9

C. 8

D. 7

E. 6
To MAXIMIZE the length of n, the prime-factorization of n must include AS MANY PRIME FACTORS AS POSSIBLE.
For this reason, we must MINIMIZE the value of each prime factor.
Since 2 is the smallest prime factor, the prime-factorization of n must include as many 2's as possible.
If n = 2*2*2*2*2*2*2*2*2 = 512, then the prime-factorization of n has nine prime factors, yielding a length of 9.
If the prime-factorization of n includes any more 2's, the value of n will exceed 1000.
Since the prime-factorization n cannot include more 2's than the 9 values in red, the greatest possible length of n = 9.

The correct answer is B.
Last edited by GMATGuruNY on Fri Sep 18, 2015 12:50 pm, edited 2 times in total.
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

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 » Fri Sep 18, 2015 7:36 am
Here's a related question to practice with: https://www.beatthegmat.com/prep-test-qu ... 78605.html

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
Image

Master | Next Rank: 500 Posts
Posts: 164
Joined: Tue May 12, 2015 9:06 am
Thanked: 3 times

by oquiella » Fri Sep 18, 2015 7:48 am
GMATGuruNY wrote:
oquiella wrote:2. What is the greatest possible length of a positive integer less than 1,000?

Note: For any positive integer n, n>1, the "length" of n is the number of positive primes whose product is n. For example, the length of 50 is 3 since 50= (2) (5) (5)


A. 10

B. 9

C. 8

D. 7

E. 6
To MAXIMIZE the length, we must MINIMIZE the value of each prime factor.
Thus, the prime-factorization of n must includes as many 2's as possible:
2*2*2*2*2*2*2*2*2 = 512.
If any other prime factors are included, the value n will exceed 1000.
Since the prime-factoization above is composed of nine 2's, the greatest possible length of n = 9.

The correct answer is B.

Why use 2? I dont understand why.

GMAT/MBA Expert

User avatar
Elite Legendary Member
Posts: 10392
Joined: Sun Jun 23, 2013 6:38 pm
Location: Palo Alto, CA
Thanked: 2867 times
Followed by:511 members
GMAT Score:800

by [email protected] » Fri Sep 18, 2015 9:45 am
Hi oquiella,

Since the question asks for the GREATEST possible "length", we need to get as many primes into the product as possible (AND keep the total less than 1,000). Consider the following:

If we used ONLY 5s....

(5)(5)(5)(5) = 625.....but this number only has a "length" of 4 (since there are only 4 primes)

If we used ONLY 3s...

(3)(3)(3)(3)(3)(3) = 729...but this number only has a "length" of 6 (since there are only 6 primes)

By using ONLY 2s, we end up with a "length" of 9 (and THAT is the greatest length possible; no matter how many other examples you might try, you won't end up with a "length" greater than 9).

GMAT assassins aren't born, they're made,
Rich
Contact Rich at [email protected]
Image

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 1462
Joined: Thu Apr 09, 2015 9:34 am
Location: New York, NY
Thanked: 39 times
Followed by:22 members

by Jeff@TargetTestPrep » Mon Dec 18, 2017 4:40 pm
oquiella wrote:2. What is the greatest possible length of a positive integer less than 1,000?

Note: For any positive integer n, n>1, the "length" of n is the number of positive primes whose product is n. For example, the length of 50 is 3 since 50= (2) (5) (5)


A. 10

B. 9

C. 8

D. 7

E. 6
In order to maximize the "length," we need to minimize the values of the prime factors of n. Since 2 is the smallest prime, let's see how many factors of 2 we can use to get a product less than 1000. Since 2^9 = 512, we see that the largest possible length of a positive integer less than 1000 is 9.

Answer: B

Jeffrey Miller
Head of GMAT Instruction
[email protected]

Image

See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews