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.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

How many prime factors does positive integer n have?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

How many prime factors does positive integer n have?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards