How many different prime factors does positive integer n have?

Question

(1) 44 < n^2 < 99
(2) 8n^2 has twelve factors

kvazar · Accepted Answer

1) only three positive integers can be derived:  7, 8, 9.  Each has only one distinct prime (7, 2 and 3 respectively). Sufficient

2) to calculate number of factors we need to find prime factorization of a number, add 1 to the powers and multiply those powers, 8 breaks to  2^3 , hence it has four factors (power of 3 + 1): 1, 2, 4, 8

We need another 2 primes to get twelve factors ( 2^3 * x^2  => ( 3+1)*(2+1) = 12 ), square root will double the primes, so we know that n has only one prime. Sufficient

Answer D

Bunuel · Answer

OFFICIAL SOLUTION: How many different prime factors does positive integer n have? (1) 44 < n^2 < 99 . This implies that n can be 7, 8, or 9. Each of these numbers has 1 prime: 7, 2, and 3, respectively. Sufficient. (2) 8n^2 has twelve factors. For 8n^2=2^3n^2 to have twelve factors n must be a prime: 8n^2=2^3*(prime)^2 --> the number of factors = (3+1)(2+1)=12 . Another case would be if n=2^4 : 8n^2=2^3*(2^4)^2=2^{11} --> the number of factors = 11+1=12 . As we see in both cases ( n = prime or n = 2^4 ), n has one prime factor. Sufficient. Answer: D

reto · Answer

Let me try to explain: If n is a positive integer: (1) 44 < n^2 < 99 This means, n can be 7, 8 or 9 > all numbers with each 1 distinct prime factor 7(7). 8(2) and 9(3). Therefore Stat. 1 Sufficient (2) 8n^2 has twelve factors 8^n2 has twelve factors. 8 on its own has 4 factors (1, 2, 4, 8), i.e. we need 8 more factors. N could be any number providing 4 factors. If n = 7, n could have 2 and 7 as prime factors which are 2 distinct factors. Therefore Stat. 2 IS. Therefore, after reviewing, I suppose it's A :oops:

Thanks for your inputs EMPOWERgmatRichC

D3N0 · Answer

Ans: D

Solution: Different prime factors of integer n? we do not know the exact value of n so finding specific value is necessery and then we can find the number of factors.
1) it gives us three values of n=7,8,9 their square  satisfy the range
 Now question asks us how many prime factors does n have
if N=7; number of prime factors =1
if N=8; number of prime factors = 1
if N=9; number of prime factors = 1
respectively.

2) 8n^2 has tweleve factors
to make is easier we write it like this (2^3)*(n^2)
 now the total number of factors for this is = 4*x=12; x=3: means n needs to be a prime number.
 so it can have any prime value. again we know as n being the prime number it has only one prime factor.

Thank You for the hint, previously i solved it for total number of factors.
Hope this is correct this time.
Ans: D

EMPOWERgmatRichC · Answer

Hi All,

You have to be VERY careful with this prompt; pay attention to the details, write EVERYTHING down and answer the question that is ASKED. Notice that the question asks how many DIFFERENT PRIME FACTORS N has....

It's a pretty safe bet that you all determined that, in Fact 1, N could be 7, 8 or 9.

How many DIFFERENT PRIME FACTORS does 7 have?
How many DIFFERENT PRIME FACTORS does 8 have?
How many DIFFERENT PRIME FACTORS does 9 have?

So what does this tell you about Fact 1?

Knowing that you have to be more detail-oriented with your work, how will you now handle Fact 2?

GMAT assassins aren't born, they're made,
Rich

thutran · Answer

How many different prime factors does positive integer n have?

(1) 44 < n^2 < 99
(2) 8n^2 has twelve factors

Solution:
(1) 44 < n^2 < 99 --> n could be 7, 8 or 9.
- 7: 1 prime factor which is 7
- 8: 1 prime factor which is 2
- 9: 1 prime factor which is 3
--> (1) is sufficient
(2)8n^2 has twelve factors:
8n^2 can be re-written as (2^3)*(n^2)
Number of factors of (2^3)*(n^2) is 12, or (3+1)*(x+1) = 12, x is the number of factor of n^2
--> n^2 has 2 factors which are n and n^2 --> n is prime number --> n has 1 prime factor
--> (2) is sufficient.

D is the answer.

kvazar · Answer

Prime factor can't have two prime factors within itself. Factors of 7 are "7" and "1". 
To check, let's put 7 in the formula from second statement:
8 * 7^2 = 2 * 2 * 2 * 7 * 7 = 2^3 * 7^2, > (3+1) * (2+1) = 12.

But you get only "sevens" from 7. 
2's come from 8, and question asks only about  n .

23a2012 · Answer

the answer is A
statment 1 suff / N^2 could be 49,64,81 and all of these numbers have one prime number
 
statment 2 insuff/ if n is prime number then n has one prime number but if not then n could has more than one prime number for example if n=3 then 8n^2=2^3*3^2 the nimber of factors is 12 and n is prime however if x=6 in this case also the number of factors will be = 12 but x has two prime factors 2,3

kvazar · Answer

Could you clarify why x=6 gives you 12 factors?

If n=6, then 8 * 6^2 = 2 * 2 * 2 * 3*2 * 3*2 = 2^5 * 3^2, hence (5+1) * (2+1) = 18 factors.

rohit8865 · Answer

Hello bunuel

Plz clear me if my approach is wrong...

As per (1) 44<n^2<99

or 4<n^2/11<9
for n to be +ve integer it should be either 5 or 7(2 primes)
so, sufficient.

reto · Answer

I am not Bunuel, but I can help. Cou have to read the question stem carefully. What is it asking? "How many different prime factors does positive integer n have?" You can answer this right away with the fact that n can either be 5 or 7 because both of these numbers have just one prime factor right?

Does it help?

Yiva · Answer

How many different prime factors does positive integer n have?

(1) 44 < n^2 < 99
(2) 8n^2 has twelve factors

I have doubt about (2). What if n is 16? then 2^3*2^8=2^11 and it has 11+1=12 factors but it has only one prime factor, which is 2. Can anyone tell me what I am missing?

reto · Answer

Statement 2 tells you that n must be prime because 8 = 2^3 which gives you (3+1) factors and for 8n^2 to have exactly 12 factors, n must be prime in order to have (3+1)(2+1). If n is prime, it has how many different PRIME Factors  Yiva ?

Yes exactly, if n is prime, n will have exactly 1 prime factor. The question can be answered.

MathRevolution · Answer

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

How many different prime factors does positive integer n have?

(1) 44 < n^2 < 99
(2) 8n^2 has twelve factors

The question has one variable (n) and 2 equations are given, so there is high chance (D) will be our answer. 
From condition 1, n^2=49,64,81, and n=7,8,9=7,2^3,3^2. There is only 1 case where there are different prime factors, so this is sufficient.
For condition 2, 8n^2=2^3n^2==>(3+1)(2+1)=12. n has to be prime. This also gives that there is only 1 case where there are different prime factors, so this is also sufficient.
The answer becomes (D).

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.

reto · Answer

What makes your approach unconventional? :-D

stonecold · Answer

Excellent Question.
Here is my solution to this one =>

We need the number of prime factors for n.

Statement 1-> 
Notice the boundary condition for n is given.
Since n is an integer => n^2 must be a perfect square.
n^2=> 49,64 or 81
So n can be 7 or 2^3 or 3^2
In each case -> n will have just one prime factor.
Hence Sufficient.

Statement 2-> 
8n^12 has twelve factors.
2^3*n^2 has 12 factors => This is only possible if n is a prime number.
Thus it will have only one prime factor.
Hence D.

LeonidK · Answer

Great question! thanks for sharing this.

Kem12 · Answer

I agree with all the explanations but have a question though... If n can be 7,8 or 9, which isn't a definite value, does that make the answer sufficient?

Posted from my mobile device

Bunuel · Answer

The question does not ask "what is the value of n", it asks "How many different prime factors does positive integer n have". So, for (1), even though n can take three values, each of those values still has only one prime factor. For (2) we get that n is a prime number, so it must have only one prime factor. Thus, both statements give that n has ONE prime factor.

How many different prime factors does positive integer n have?

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GMAT Data Sufficiency (DS) Questions

How many different prime factors does positive integer n have?

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