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 Post subject: GMAT Number Theory
PostPosted: Thu Oct 28, 2010 5:06 pm 
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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.


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


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