If a natural number p has 8 factors, then which of the

Question

If a natural number p has 8 factors, then which of the following cannot be the difference between the number of factors of p^3 and p?

A. 14
B. 30
C. 32
D. 56
E. None of these

Bunuel · Accepted Answer

Finding the Number of Factors of an Integer

First make prime factorization of an integer  n=a^p*b^q*c^r , where  a ,  b , and  c  are prime factors of  n  and  p ,  q , and  r  are their powers.
 
The number of factors of  n  will be expressed by the formula  (p+1)(q+1)(r+1) .  NOTE:  this will include 1 and n itself.

Example:  Finding the number of all factors of 450:  450=2^1*3^2*5^2

Total number of factors of 450 including 1 and 450 itself is  (1+1)*(2+1)*(2+1)=2*3*3=18  factors.

BACK OT THE ORIGINAL QUESTION:

We are told that  p  has 8 factors: 8=2*4=2*2*2.

If  p  has only one prime in its prime factorization, say  a , then  p=a^7  --> number of factors of  p  is  (7+1)=8  -->  p^3=(a^7)^3=a^{21}  in this case would have  (21+1)=22  factors --> the difference is  22-8=14 ;

If  p  has two primes in its prime factorization, say  a  and  b , then:  p=a^3*b  --> number of factors of  p  is  (3+1)(1+1)=8  -->  p^3=a^9*b^3  in this case would have  (9+1)(3+1)=40  factors --> the difference is  40-8=32 ;

If  p  has three primes in its prime factorization, say  a ,  b  and  c , then:  p=a*b*c  --> number of factors of  p  is  (1+1)(1+1)(1+1)=8  -->  p^3=a^3*b^3*c^3  in this case would have  (3+1)(3+1)(3+1)=64  factors --> the difference is  64-8=56 .

p cannot have more than 3 factors, since the least number of factors a number with four primes can have is 16>8 (for example if  p=abcd , then number of factors of  p  is  (1+1)(1+1)(1+1)(1+1)=16 ).

Answer: B.

Vips0000 · Answer

p has 8 factors. That gives following possiblities:

a) p has one prime factor with power 7
=> p^3 will have 22 factors. Diffrerence with number of factors of p = 22-8 = 14: A is ok
b) p has two prime factors with powers 3 and 1
=> p^3 will have 40 factors. Difference with number of facors of p =40-8 = 32 : C is ok
c) p has three prime factors with powers 1 each
=> p^3 will have 64 factors. Difference with number of factors of p=64-8 = 56 : D is ok

There are no other possiblities. Hence remaining answer choice B is not possible.

Ans B it is!

Amateur · Answer

i didnot understand a bit of your explanation.... can you please explain it? thank you

seths22 · Answer

Since p has 8 total factors, we can assume that the prime factorization of p will take three forms based on the principle that the total number of factors of an integer can be derived from the prime factorization of that integer where the integer's total factors = prime factors (a +1) * (b +1):

p --> x^7 (where 7+1 = 8 total factors)

p -->x^3 * y^1 (or y^3 * x^1) (where (3+1)*(1+1) = 8 total factors)

p -->x^1 * y^1 * z^1 (where (1+1) * (1+1) * (1 +1) = 8 total factors)

Now, we can address the possible total factors of (p^3 - p) as follows:

(x^7)^3 = (x)^21 + 1 = 22 total factors --> 22 - 8 =  14

(x^3)^3 * (y^1)^3 = (9+1) * (3+1) --> 40 - 8 =  32

(x^1)^3 * (y^1)^3 * (z^1)^3 = (3+1) * (3+1) * (3+1) = 64 - 8 =  56

Thus,  (B)  is the only answer that is not possible.

Great question. Source?

Aristocrat · Answer

Good explanation bunuel.
I was calculating for 3p and p.
The way question is posted

Bunuel · Answer

From hardest questions set.

Bumping for review and further discussion.

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.

If a natural number p has 8 factors, then which of the

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 Problem Solving (PS) Questions

If a natural number p has 8 factors, then which of the

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