How many prime factors does positive integer n have?

Question

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

Zhenek · Accepted Answer

#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

Bunuel · Answer

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.

Answer (E)

Harley1980 · Answer

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.

Answer is A

aniteshgmat1101 · Answer

Answer should be B.
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.

hsbinfy · Answer

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

andreinspb · Answer

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.

tigabalm · Answer

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?

Bunuel · Answer

It means one distinct prime factor.

stonecold · Answer

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

carlospascu · Answer

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

Bunuel · Answer

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).

ShankSouljaBoi · Answer

Statement 1 : Insufficient --- n could be 7 or could be a multiple of 7 and 3
Statement 1 : Insufficient --- n could be 7 or could be a multiple of 7 and 3
Combining : No new info
E it is

Ahmed9955 · Answer

Can someone help me understand 
how to take up the problem when I combine S1 and S2

phambrenda · Answer

Question: How many primes does "n" have?

1) n/7 = one prime

- 49/7 = 7 one prime number (7)
-21/7= 3 two prime numbers (7,3)

2) 3*n^2 has two different prime factors.

- 3*6^2 = two prime factors but n itself has 2 primes (3,2)
- 3*2^2 = two prime factors but n itself has 1 prime number (2)

NS for both

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How many prime factors does positive integer n have?

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