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# How many prime factors does N have?

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How many prime factors does N have?  [#permalink]

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Updated on: 29 Oct 2018, 21:47
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2
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Difficulty:

75% (hard)

Question Stats:

38% (01:28) correct 63% (01:42) wrong based on 119 sessions

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How many prime factors does N have?

(1) The least prime factor of N is greater than √N.

(2) N has 2 positive factors.

Originally posted by piyush26 on 29 Oct 2018, 11:21.
Last edited by Bunuel on 29 Oct 2018, 21:47, edited 1 time in total.
Renamed the topic.
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Re: How many prime factors does N have?  [#permalink]

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29 Oct 2018, 15:19
2
piyush26 wrote:
How many prime factors does the positive integer N have?

(1) The least prime factor of N is greater than √N.

(2) N has exactly 2 positive factors.

Very nice conceptual problem, piyush26 (kudos) !

$$?\,\, = \,\,\,\,\# \,\,{\text{prime}}\,{\text{factors}}\,\,{\text{of}}\,\,N\,\,\,\,\,\,\left( {N \geqslant 1\,\,\operatorname{int} } \right)$$

$$\left( 1 \right)\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,N\,\,{\text{prime}}\,\,\,\,\, \Rightarrow \,\,\,\,? = 1\,\,\,\left( {N\,\,{\text{itself}}} \right)$$

$$\left( * \right)\,\,\left\{ \begin{gathered} N = 1\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{no}}\,\,{\text{prime}}\,\,{\text{factors}} \hfill \\ N \geqslant 4\,\,{\text{not}}\,\,{\text{prime}}\,\,\,\,\, \Rightarrow \,\,\,\,\,\,2 \leqslant p\,\,{\text{prime}}\,\,{\text{factor}}\,\,{\text{of}}\,\,N\,\, \leqslant \sqrt N \,\,{\text{exists}} \hfill \\ \end{gathered} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{contradicts}}\,\,\left( 1 \right)$$

$$\left( 2 \right)\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,N\,\,{\text{prime}}\,\,\,\,\, \Rightarrow \,\,\,\,? = 1\,\,\,\left( {N\,\,{\text{itself}}} \right)$$

$$\left( {**} \right)\,\,\left\{ \begin{gathered} N = 1\,\,\,\,\, \Rightarrow \,\,\,{\text{only}}\,\,1\,\,{\text{positive}}\,\,{\text{factor}}\,\,\left( {N\,\,{\text{itself}} } \right) \hfill \\ N \geqslant 4\,\,{\text{not}}\,\,{\text{prime}}\,\,\,\,\, \Rightarrow \,\,\,{\text{more}}\,\,{\text{than}}\,\,{\text{the}}\,\,{\text{trivial}}\,\,{\text{two}}\,\,\left( {1\,\,{\text{and}}\,\,N} \right)\,\,{\text{positive}}\,\,{\text{factors}} \hfill \\ \end{gathered} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{contradicts}}\,\,\left( 2 \right)\,\,\,$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: How many prime factors does N have?  [#permalink]

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29 Oct 2018, 18:11
Hii
Statement 1: least prime factor of N is greater than √N, hence no there is no factor of N except 1 and the number N inself. It is a prime number. So no of prime factors =1. Sufficient.

Statement 2:
Since N has 2 prime factors, it is a prime number and hence the no of prime factors is 1. Sufficient.

Don't forget to tag me in case of any doubt.

piyush26 wrote:
How many prime factors does N have?

(1) The least prime factor of N is greater than √N.

(2) N has 2 positive factors.

Posted from my mobile device
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Re: How many prime factors does N have?  [#permalink]

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29 Oct 2018, 19:48
Why statement 1 is sufficient? is because of 2?
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Re: How many prime factors does N have?  [#permalink]

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29 Oct 2018, 20:00
1
we need to check for factors only upto $$\sqrt{n}$$ because the factors repeat after this. See attached file
Attachment:

2018_10_30 08_26 Office Lens.jpg [ 98.8 KiB | Viewed 796 times ]

jorgetomas9 wrote:
Why statement 1 is sufficient? is because of 2?

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Re: How many prime factors does N have?  [#permalink]

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30 Oct 2018, 10:27
piyush26 wrote:
How many prime factors does N have?

(1) The least prime factor of N is greater than √N.

(2) N has 2 positive factors.

Stmt 2 is obviously sufficient

let's look at stmt 1:

In GMAT world we don't have sq rt of -ve numbers ... thus we will consider only whole numbers.

0 and 1 are out of commission because they don't have prime factors.

2 --> least prime factor pf 2 is 2 which is greater than sq rt of 2 i.e. 1.41
3 and other prime numbers --> same as 2
4 --> least prime factor = sq rt of 4 = 2 .... this doesn't satisfy our condition ... thus we can't consider 4

For all even numbers > 4, least prime factor = 2 will be lesser than sq rt of that number thus we can't consider those numbers

9 --> least prime factor = sq rt of 9 = 3 .... this doesn't satisfy our condition ... thus we can't consider 9

Next odd non prime number = 15 --> least prime number = 3 which is lesser than sq rt of 15 .... thus we can't consider odd non prime numbers ....

This leaves us with ONLY PRIME NUMBERS.

This is sufficient to answer our question.

I hope that helps!
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Re: How many prime factors does N have?  [#permalink]

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30 Oct 2018, 13:11
piyush26 wrote:
How many prime factors does N have?

(1) The least prime factor of N is greater than √N.

(2) N has 2 positive factors.

Either statement suggests that N is prime.

D
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Re: How many prime factors does N have?  [#permalink]

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01 Nov 2018, 22:25
How many prime factors does N have?

(1) The least prime factor of N is greater than √N.

(2) N has 2 positive factors.

1. Imagine, this is possible only when the greatest prime factor is the prime number itself. Hence, N is a prime number and has only 1 prime factor- N itself.

2. This is possible only for a prime number. Only a prime number has 2 factors. Any prime number has one prime factor (the number itself). Hence, the answer is 1. Sufficient.

D.
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Re: How many prime factors does N have?  [#permalink]

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02 Nov 2018, 04:12
Aamirso wrote:
piyush26 wrote:
How many prime factors does N have?

(1) The least prime factor of N is greater than √N.

(2) N has 2 positive factors.

Either statement suggests that N is prime.

D

Hi,

Please try to post meaningful posts. The option D in DS questions means that both the statements are individually sufficient to answer the question.

Best.

Posted from my mobile device
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Re: How many prime factors does N have?   [#permalink] 02 Nov 2018, 04:12
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