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Post subject: GMAT Number Theory Posted: Thu Sep 02, 2010 1:33 am 

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 5element 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: 39 (1, 3, 13, 39); 5: 35 (1, 5, 7, 35); 6: 33 (1, 3, 11, 33). 
In this case, cube of 3 is 27 and if you consider that its distinct odd factors are 3 and 9. Unless 1 and 27 is not included, there are not 4 distinct factors. But 1 and 27 are not generally considered factors. If so, then every prime number will have a factor. Am i missing something here?


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Gennadiy

Post subject: Re: math (test 1, question 23): numbers theory Posted: Thu Sep 02, 2010 2:00 am 

Joined: Sun May 30, 2010 2:23 am Posts: 498

When we count a number of distinct factors of n we count ALL the possible divisors of a number n. 1 and n itself are also divisors. Let us look at definition.
Definition: A factor of n is an integer which divides n without leaving a remainder.
Note, that any positive integer n, greater than 1, has at least two factors: 1 and n itself. Any prime number is also divisible by 1 and itself.
Definition: A prime number is a positive integer that has exactly two distinct postivie integer divisors: 1 and itself.
Therefore never forget 1 and n itself when counting factors of n.
EXAMPLES: Factors of 12 are: 1, 2, 3, 4, 6, 12. Factors of 7 are: 1, 7. Factors of 27 are: 1, 3, 9, 27. Factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30. Factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
We have a different situation when we count number of distinct PRIME factors. In this case 1 is not a PRIME factor. If n is not prime then n itself is not a PRIME factor of n as well.
EXAMPLES: PRIME factors of 12 are: 2, 3. PRIME factors of 7 are: 7. PRIME factors of 27 are: 3. PRIME factors of 30 are: 2, 3, 5. PRIME factors of 48 are: 2, 3.


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