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Manager  Joined: 24 Aug 2012
Posts: 100
If a natural number p has 8 factors, then which of the  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 54% (02:47) correct 46% (02:16) wrong based on 248 sessions

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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
Math Expert V
Joined: 02 Sep 2009
Posts: 60647
Re: If a natural number ‘p’ has 8 factors, then which of the fol  [#permalink]

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13
3
Amateur wrote:
Vips0000 wrote:
kingb wrote:
If a natural number ‘p’ has 8 factors, then which of the following cannot be the difference between the number of factors of p3 and p
a. 14
b. 30
c. 32
d. 56
e. None of these

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!

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

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 $$(1+1)(1+1)(1+1)(1+1)=16$$).

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Re: If a natural number ‘p’ has 8 factors, then which of the fol  [#permalink]

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4
3
kingb wrote:
If a natural number ‘p’ has 8 factors, then which of the following cannot be the difference between the number of factors of p3 and p
a. 14
b. 30
c. 32
d. 56
e. None of these

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!
##### General Discussion
Manager  Joined: 05 Nov 2012
Posts: 138
Re: If a natural number ‘p’ has 8 factors, then which of the fol  [#permalink]

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Vips0000 wrote:
kingb wrote:
If a natural number ‘p’ has 8 factors, then which of the following cannot be the difference between the number of factors of p3 and p
a. 14
b. 30
c. 32
d. 56
e. None of these

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!

i didnot understand a bit of your explanation.... can you please explain it? thank you
Intern  Joined: 18 Jul 2010
Posts: 4
Re: If a natural number ‘p’ has 8 factors, then which of the fol  [#permalink]

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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?
Manager  Joined: 29 Jul 2012
Posts: 130
GMAT Date: 11-18-2012
Re: If a natural number ‘p’ has 8 factors, then which of the fol  [#permalink]

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Good explanation bunuel.
I was calculating for 3p and p.
The way question is posted
Math Expert V
Joined: 02 Sep 2009
Posts: 60647
Re: If a natural number p has 8 factors, then which of the  [#permalink]

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From hardest questions set.

Bumping for review and further discussion.
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Re: If a natural number p has 8 factors, then which of the  [#permalink]

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_________________ Re: If a natural number p has 8 factors, then which of the   [#permalink] 02 Dec 2019, 13:02
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