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How many odd positive divisors does 9000 have?

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BrentGMATPrepNow
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How many odd positive divisors does 9000 have? [#permalink] New post   12 Feb 2017, 07:53
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69% (01:12) correct 31% (01:53) wrong based on 362 sessions
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How many odd positive divisors does 9000 have?

A) 6
B) 8
C) 10
D) 12
E) 15

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D
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Re: How many odd positive divisors does 9000 have? [#permalink] New post   12 Feb 2017, 08:59
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first find out prime factors of 9000

that is 3^2*5^3*2^3 . so we have 48 distinct factors. we need odd. We can only get odd by multiplying odd factors. so we must use only 5^3*3^2.
from that we can calculate odd distinct factors: (3+1)*(2+1)=12


answer is D
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Re: How many odd positive divisors does 9000 have? [#permalink] New post   12 Feb 2017, 10:02
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GMATPrepNow wrote:
How many odd positive divisors does 9000 have?

A) 6
B) 8
C) 10
D) 12
E) 15

* kudos for all correct solutions


IMPORTANT RULE
If the prime factorization of N = (p^a)(q^b)(r^c) . . . (where p, q, r, etc are different prime numbers), then N has a total of (a+1)(b+1)(c+1)(etc) positive divisors.

Example: 14000 = (2^4)(5^3)(7^1)
So, the number of positive divisors of 14000 = (4+1)(3+1)(1+1) =(5)(4)(2) = 40

9000 = (2)(2)(2)(3)(3)(5)(5)(5)
= (2³)(3²)(5³)

We want ODD divisors only, so we need to disregard the 2's, because they will give us EVEN divisors.
In other words, if we examine the divisors of (3²)(5³), we will find that ALL of them are ODD (since any product of odd integers will be odd)
The number of positive divisors of (3²)(5³) = (2+1)(3+1) =(3)(4) = 12

Answer: D

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Re: How many odd positive divisors does 9000 have? [#permalink] New post   13 Feb 2017, 04:00
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GMATPrepNow wrote:
How many odd positive divisors does 9000 have?

A) 6
B) 8
C) 10
D) 12
E) 15

* kudos for all correct solutions


9000 = 2^3 * 3 ^2 * 5^3
we need only odd factor ,so rejecting all the factor containing 2 as one of the factor..
odd factor - (2+1)(3+1)
12.
D.
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Re: How many odd positive divisors does 9000 have? [#permalink] New post   13 Feb 2017, 09:53
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GMATPrepNow wrote:
How many odd positive divisors does 9000 have?

A) 6
B) 8
C) 10
D) 12
E) 15

* kudos for all correct solutions


\(9000 = 2^3 * 3 ^2 * 5^3\)

The trick here is considering all the factors except 2 ( Because 2 will always create an even number )

Hence, the required number of odd Positive is ( 3 + 1 )( 2 + 1 ) = 12

Answer will be (D) 12
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Re: How many odd positive divisors does 9000 have? [#permalink] New post   20 Apr 2024, 06:47
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Re: How many odd positive divisors does 9000 have? [#permalink]
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