How many prime factors does positive integer n have?

Question

(1) n/5 has only a prime factor.
(2) 3*n^2 has two different prime factors.

shrouded1 · Answer

(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)

subhashghosh · Answer

(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.

stonecold · Answer

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

EddyEd · Answer

I understand perfectly why (1) and (2) separately are insufficient, but I'm stuck at analyzing both statements together. Can anyone shed some light?

hellosanthosh2k2 · Answer

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 !

bkastan · Answer

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?

danielct · Answer

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?

Bunuel · Answer

1 is NOT a prime number.

2. Properties of Integers  
     Theory   
 Math: Number Theory 
 Number Properties: Tips and hints 
 Advanced Number Properties on GMAT (All 5 parts) 
 Do you make these 3 mistakes in GMAT Even-Odd Questions?  
 Everything about Factorials on the GMAT 
 Solving Complex Problems Using Number Line 
     Questions   
 Tough New Set: Number Properties Set 
 Power of a Number in a Factorial Problems 
 Number Properties Ability Quiz 
 Special Numbers and Sequences Problems 
 The E-GMAT Number Properties Knockout: 10 Great 700+ Ques for you! 
 Trailing Zeros Problems 
 DS Questions 
 PS Questions

For more check  Ultimate GMAT Quantitative Megathread

Hope it helps.

fskilnik · Answer

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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How many prime factors does positive integer n have?

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