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How many prime factors does positive integer n have?

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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.
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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

Hence, Answer is (e)
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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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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?
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How many prime factors does positive integer n have?  [#permalink]

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14 Nov 2017, 12:35
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Ed_Palencia wrote:
I understand perfectly why (1) and (2) separately are insufficient, but I'm stuck at analyzing both statements together. Can anyone shed some light?

Hi Ed_Palencia

Statement 1 => implies, n can have two factors eg $$(3 * 5)$$
$$or$$
can have only one prime factor, but $$n = 5 * 5$$, so that $$n/5$$ has one prime factor => insuff

Statement 2 => implies, $$3n^2$$=> two diff prime factors, so $$n$$ can be $$(3 * 5) or (5 * 5)$$
if $$n = 3 * 5$$,
$$3n^2$$ => $$3(3 * 5)$$ => two prime factors
if $$n = 5 * 5$$,
$$3n^2$$ => $$3(5 * 5)$$ => two prime factors

so if you combine, still n can be $$3 * 5 or 5 * 5$$, not sufficient to tell how many prime factors

Answer (E)

Hope that helps !
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How many prime factors does positive integer n have?  [#permalink]

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19 Nov 2017, 10:01
shrouded1 wrote:
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

Hence, Answer is (e)

shrouded1

For (1) isn't the statement saying that n/5 has only 1 prime factor? i.e: how can our tests show that it has 1 or 2 prime factors if statement says n/5 has only 1 prime factor?
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Re: How many prime factors does positive integer n have?  [#permalink]

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22 Nov 2017, 14:42
Hi guys, I haven't understood why 1) is insufficient. My thought is:
if n/5 has only a prime factor, n/5 must be equal to 1,thus n = 5.

if n were 25(or any 5^x), n/5 would have 2 prime factors : 5 and 1.

Could you explain what I've missed?
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Re: How many prime factors does positive integer n have?  [#permalink]

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23 Nov 2017, 00:40
danielct wrote:
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.

Hi guys, I haven't understood why 1) is insufficient. My thought is:
if n/5 has only a prime factor, n/5 must be equal to 1,thus n = 5.

if n were 25(or any 5^x), n/5 would have 2 prime factors : 5 and 1.

Could you explain what I've missed?

1 is NOT a prime number.

2. Properties of Integers

For more check Ultimate GMAT Quantitative Megathread

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

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03 Oct 2018, 06:47
banksy wrote:
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.

VERY nice one!

$$n \ge 1\,\,{\mathop{\rm int}}$$

$$?\,\, = \,\,\# \,\,{\rm{prime}}\,\,{\rm{factors}}\,\,{\rm{of}}\,\,n$$

$$\left( 1 \right)\,\,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,n = {5^2}\,\,\,\, \Rightarrow \,\,\,\,\,? = 1\,\,\,\,\,\,\,\,\,\,\left( {{\rm{just}}\,\,5} \right)\, \hfill \cr \,{\rm{Take}}\,\,n = 5 \cdot 2\,\,\,\, \Rightarrow \,\,\,\,\,? = 2\,\,\,\,\,\,\,\left( {2\,\,{\rm{and}}\,\,5} \right) \hfill \cr} \right.$$

$$\left( 2 \right)\,\,\,\,\left\{ \matrix{ \,({\rm{Re}})\,{\rm{Take}}\,\,n = {5^2}\,\,\,\left( {3 \cdot {5^4}\,\,{\rm{has}}\,\,{\rm{just}}\,\,3\,\,{\rm{and}}\,\,5} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = 1\,\,\,\,\left( {{\rm{just}}\,\,5} \right)\,\,\,\,\,\,\,\,\, \hfill \cr \,{\rm{Take}}\,\,n = 2 \cdot 3\,\,\,\left( {{3^3} \cdot {2^2}\,\,{\rm{has}}\,\,{\rm{just}}\,\,3\,\,{\rm{and}}\,\,2} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 2\,\,\,\,\,\left( {2\,\,{\rm{and}}\,\,3} \right)\,\,\,\,\,\,\, \hfill \cr} \right.$$

$$\left( {1 + 2} \right)\,\,\,\left\{ \matrix{ \,({\rm{Re}})\,{\rm{Take}}\,\,n = {5^2}\,\,\,\left( {3 \cdot {5^4}\,\,{\rm{has}}\,\,{\rm{just}}\,\,3\,\,{\rm{and}}\,\,5} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = 1\,\,\,\,\left( {{\rm{just}}\,\,5} \right)\,\,\,\,\,\,\,\,\, \hfill \cr \,{\rm{Take}}\,\,n = 3 \cdot 5\,\,\,\left( {{3^3} \cdot {5^2}\,\,{\rm{has}}\,\,{\rm{just}}\,\,3\,\,{\rm{and}}\,\,5} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 2\,\,\,\,\,\left( {3\,\,{\rm{and}}\,\,5} \right)\,\,\,\,\,\,\, \hfill \cr} \right.$$

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

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

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Re: How many prime factors does positive integer n have?   [#permalink] 15 Dec 2019, 02:06
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