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Math Expert V
Joined: 02 Sep 2009
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How many prime factors does positive integer n have?  [#permalink]

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21 00:00

Difficulty:   95% (hard)

Question Stats: 18% (02:08) correct 82% (01:54) wrong based on 234 sessions

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How many prime factors does positive integer n have?

(1) n/7 has only one prime factor.
(2) 3*n^2 has two different prime factors.

Kudos for a correct solution.

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Math Expert V
Joined: 02 Sep 2009
Posts: 59725
Re: How many prime factors does positive integer n have?  [#permalink]

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Bunuel wrote:
How many prime factors does positive integer n have?

(1) n/7 has only one prime factor.
(2) 3*n^2 has two different prime factors.

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:

Let’s keep in mind our learning from above while trying to solve this question.

Statement 1: n/7 has only one prime factor.

n/7 has a factor so obviously, it is an integer. Hence n must have a 7 as a factor. So we might jump to the conclusion that n has two prime factors –7 and another one which is left when n is divided off by 7. So n would be something like 7*3 so that n/7 = 7*3/7 = 3 (only one prime factor).

But what we wouldn’t have considered in this case is that n may have multiple 7s so that when a 7 is cancelled in n/7, you would still be left with 7 i.e. if n is 7*7, then n/7 = 7*7/7 = 7. In this case, n has only one distinct prime factor.

So n can have either one or two prime factors. This statement alone is not sufficient.

Statement 2: 3*n^2 has two different prime factors.

This is the same as our previous question. 3n^2 has two different prime factors but n itself can have either one or two prime factors (one of which will be 3). For example, n can be 7 or n can be 3*7. This statement alone is not sufficient.

Using both statements, n could have one or two prime factors i.e. n could be 49 (only one prime factor – 7) or n could be 21 (two prime factors).

Hence, even using both the statements, we cannot say how many prime factors n has.

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

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8
2
#1 - n/7 has only 1 prime factor.
1: n = 49: 49/7 = 7. 1 prime factor, n has 1 prime factor
2: n = 21: 21/7 = 3. 1 prime factor, n has 2 prime factors

Insufficient

#2 - 3*n^2 - has 2 different prime factors
1: n = 2: 3*4 = 12, 2 and 3 are 2 different prime factors, n has 1 prime factor
2: n = 6: 3*6*6 = 3*3*2*6. 2 and 3 - 2 different prime factors, n has 2 prime factors (2 and 3)
Insufficient

their combination doesn't yield any result either.
from #1 we get n = 7k where k = prime, so $$3*7*7*k^2$$
k can be 3 or 7, which would give us n = 21 and 49, which have 2 and 1 prime factors respectively

E

Originally posted by Zhenek on 07 Apr 2015, 03:02.
Last edited by Zhenek on 08 Apr 2015, 03:31, edited 1 time in total.
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Senior Manager  Joined: 02 Mar 2012
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Re: How many prime factors does positive integer n have?  [#permalink]

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1
E for me too.

1) n can be 3*7 or 7*7

2) n can be 1 prime number or multiple of 3 and a prime number
n can be 7 or 21 taken as an example..both of the cases 7 and 21 have two prime factors

takeing both also we dont have solution.

So E
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GMAT 1: 660 Q48 V33 GMAT 2: 740 Q50 V40 Re: How many prime factors does positive integer n have?  [#permalink]

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Bunuel wrote:
How many prime factors does positive integer n have?

(1) n/7 has only one prime factor.
(2) 3*n^2 has two different prime factors.

Kudos for a correct solution.

1) n have two prime factors: 7 and some another prime factor that remains when we divide n by 7. Sufficient

2) $$3*n^2$$ . In this statement we have two possible variants n have factor 3 than n have two prime factors and n doesn't have prime factor 3 than n have only 1 prime factor. Insufficient.

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

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1
1) In both cases: n = 7 * 7 and n = 7 * 3, n/7 has only one prime factor, but number of prime factors of n vary (1 and 2 respectively). INSUFFICIENT.

2) In 3*n^2 has 2 prime factors, one prime factor is 3 for sure and another prime factor definitely is coming from n^2.
Which means even n should have only one prime factor. SUFFICIENT.
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Re: How many prime factors does positive integer n have?  [#permalink]

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There is no word 'distinct' in the task, so the number 49 when factored has two primes in its decomposition, 7 and 7, but only one distinct prime 7. Depending on the word 'distinct' answer can be D or E.
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Re: How many prime factors does positive integer n have?  [#permalink]

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Sorry for the late bump.

I am wondering though because st.1 doesn't say that n/7 has one distinct prime factor, it only says that it has 1 prime factor. Are we to take this as it can mean 1 or 1 distinct?
Math Expert V
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Re: How many prime factors does positive integer n have?  [#permalink]

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tigabalm wrote:
Sorry for the late bump.

I am wondering though because st.1 doesn't say that n/7 has one distinct prime factor, it only says that it has 1 prime factor. Are we to take this as it can mean 1 or 1 distinct?

It means one distinct prime factor.
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GRE 1: Q169 V154 How many prime factors does positive integer n have?  [#permalink]

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Excellent Question.
Here is what i did solution -->

Given data -->
n is a positive integer.
We need the number of prime factors of n.

Statement 1-->
n/7 = one prime factor.
n=2*7 => Acceptable
n=7^3 =>Acceptable

Hene n can have either one or two prime factors.
Hence Not sufficient.

Statement 2 -->
3*n^2 has two primes.
n=7^2 => Acceptable
n=3*7=> Acceptable

Hence n can have either one or two prime factors.

Hence insufficient.

Combing the two statements
n=7^2 => Acceptable
n=3*7 => Acceptable

Hence n can have one or two primes.

Hence insufficient

Hence E

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

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[quote="Bunuel"]How many prime factors does positive integer n have?

(1) n/7 has only one prime factor.
(2) 3*n^2 has two different prime factors.

I think that the answer is not well written becasue, you could interpret it in two ways:
1) How many prime factors does positive integer n have? -- the answer is 1 primer factor integer n has -- So Ans D
2) How many possible prime factors does positive integer n have? -- the answer would be E
Math Expert V
Joined: 02 Sep 2009
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Re: How many prime factors does positive integer n have?  [#permalink]

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carlospascu wrote:
Bunuel wrote:
How many prime factors does positive integer n have?

(1) n/7 has only one prime factor.
(2) 3*n^2 has two different prime factors.

I think that the answer is not well written becasue, you could interpret it in two ways:
1) How many prime factors does positive integer n have? -- the answer is 1 primer factor integer n has -- So Ans D
2) How many possible prime factors does positive integer n have? -- the answer would be E

n is some specific number. Using both statements, n could have one or two prime factors i.e. n could be 49 (only one prime factor – 7) or n could be 21 (two prime factors).
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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] 22 Apr 2019, 08:56
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