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# How many prime factors does positive integer n have?

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Manager
Joined: 10 Feb 2011
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How many prime factors does positive integer n have? [#permalink]

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29 Mar 2011, 12:30
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How many prime factors does positive integer n have?

(1) n/5 has only a prime factor.
(2) 3*n^2 has two different prime factors.
[Reveal] Spoiler: OA

Kudos [?]: 363 [2], given: 10

Retired Moderator
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Re: How many prime factors does positive integer n have? [#permalink]

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29 Mar 2011, 13:54
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banksy wrote:
49. How many prime factors does positive integer n have?
(1) n/5 has only a prime factor.
(2) 3*n^2 has two different prime factors.

(1) : n/5 has one prime factor. So we know immediately that is a multiple of 5. Either n is of the form 5^k or 5x(another_prime)^k (Eg. n=125 or n=15 both work). Hence n has either 1 or 2 prime factors ... Insufficient

(2) : 3*n^2 has two factors. Again, n could have 1 or 2 factors. Eg n=15 OR n =125 both work

(1+2) : Take the case n = 15 and n = 125 ... both statements can be true together. Hence not clear if n has one prime factor or two

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Re: How many prime factors does positive integer n have? [#permalink]

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29 Mar 2011, 18:00
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(1) n/5 has one prime factor. So n could be 25 or 15, in which case n/5 = 5 or n/5 = 3, so n can have more than one prime factor. So Insufficient.

(2) 3*n^2 has two different prime factors. So if n = 5*3 = 15, or n = 5 or 25, then also the expression has two distinct prime factors. So Insufficient.

In (1) and (2), 15 can work, or 25 can work, so answer is E.
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Re: How many prime factors does positive integer n have? [#permalink]

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17 Mar 2015, 22:23
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Re: How many prime factors does positive integer n have? [#permalink]

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08 Dec 2016, 00:05
Excellent Question.
Here is what i did in this one =
n=3*5 and 5^3 satisfy both the equations
Hence n can have one or two prime factors.

Hence E

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Re: How many prime factors does positive integer n have? [#permalink]

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23 Sep 2017, 07:47
I understand perfectly why (1) and (2) separately are insufficient, but I'm stuck at analyzing both statements together. Can anyone shed some light?

Kudos [?]: 7 [0], given: 60

Re: How many prime factors does positive integer n have?   [#permalink] 23 Sep 2017, 07:47
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