It is currently Sun Oct 21, 2018 8:29 pm

 All times are UTC - 5 hours [ DST ]

 Page 1 of 1 [ 2 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: GMAT Number TheoryPosted: Thu Oct 28, 2010 5:06 pm

Joined: Sun May 30, 2010 3:15 am
Posts: 424
How many positive integers less than 50 have exactly 4 distinct odd factors but no even factors?
A. 6
B. 7
C. 8
D. 9
E. 10

(A) Any positive integer with exactly 4 odd factors must be either:(i) a product of two odd and distinct prime numbers;or (ii) the cube of an odd prime number.

Let’s look at (i): the two distinct prime factors of the number we want must be members of the 5-element set {3, 5, 7, 11, 13}.
3 could be paired with any of the 4 other elements.
5 can be paired with only with one element, number 7.
Add up the possibilities for 3 and 5 and you get 5 products of distinct odd prime numbers that are less than 50.

Turning to (ii), only the cube of 3 will be less than 50.

In all then, there are 6 positive integers less than 50 that have 4 odd factors and no even ones. The right answer is choice (A).

Another way to tackle this problem is just to find all the numbers (ideally within less than 3 minutes).
1: 15 (1, 3, 5, 15);
2: 21 (1, 3, 7, 21);
3: 27 (1, 3, 9, 27);
4: 33 (1, 3, 11, 33);
5: 35 (1, 5, 7, 35);
6: 39 (1, 3, 13, 39);
-------------

Can 45 be one of these numbers? It can be divided by 1, 45, 5, and 9.

Top

 Post subject: Re: math (test 1, question 23): number theory.Posted: Thu Oct 28, 2010 5:31 pm

Joined: Sun May 30, 2010 2:23 am
Posts: 498
No, 45 is NOT one of those numbers.
The divisors of 45 are: 1, 3, 5, 9, 15, 45. (6 distinct odd divisors).

The idea of this question is following:
- Each integer is divisible by itself and 1.
- If an integer is even then it has at least one even factor (itself).
- If an integer is odd then it has at least two odd factors (itself and 1).
- If an integer is prime then it has exactly two factors (itself and 1).
- If an integer n is not prime then it has at least one prime factor a. The integer has one more distinct factor n/a, if n/a does not equal a. In order for the integer to have exactly 4 distinct, factors n/a must be either a prime number or a².

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 2 posts ]

 All times are UTC - 5 hours [ DST ]

#### Who is online

Users browsing this forum: No registered users and 3 guests

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forumYou cannot post attachments in this forum

Search for:
 Jump to:  Select a forum ------------------ GMAT    GMAT: Quantitative Section (Math)    GMAT: Verbal Section    GMAT: Integrated Reasoning    GMAT: General Questions GRE    GRE: Quantitative Reasoning (Math)    GRE: Verbal Reasoning    GRE: General Questions General questions    Other questions