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# If p is a positive integer, what is total number of positive factors

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Math Expert
Joined: 02 Sep 2009
Posts: 61537
If p is a positive integer, what is total number of positive factors  [#permalink]

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16 Dec 2019, 00:35
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If p is a positive integer, what is total number of positive factors of 10p

(1) p has total of 3 positive factors

(2) The greatest common factor of 10 and p is 1

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Re: If p is a positive integer, what is total number of positive factors  [#permalink]

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16 Dec 2019, 01:55
#1
p has total of 3 positive factors

square of a prime no always has 3 factors
so p is prime ; factors of prime are 2 and 10 ; 2^1*5^1 ; 4 ; total + factors of 10p ; 6
sufficient
#2
The greatest common factor of 10 and p is 1

p is prime ; sufficient
IMO D

Bunuel wrote:
If p is a positive integer, what is total number of positive factors of 10p

(1) p has total of 3 positive factors

(2) The greatest common factor of 10 and p is 1

Are You Up For the Challenge: 700 Level Questions
Intern
Joined: 20 Dec 2019
Posts: 35
Re: If p is a positive integer, what is total number of positive factors  [#permalink]

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27 Dec 2019, 10:58
Here the question is not about different positive factors, its asking only positive factors - ie they might repeat.
Hence the answer should be A
VP
Joined: 19 Oct 2018
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Location: India
Re: If p is a positive integer, what is total number of positive factors  [#permalink]

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27 Dec 2019, 12:21
1
Statement 1-

p has total 3 positive factors. Hence p is square of a prime number

Case 1- p=2^2 or 5^2

Positive factors of 10 p = 4*2=8

Case 2- p is square of prime number other than 2 or 5

Positive factors of 10 p= 2*2*3= 12

Insufficient

Statement 2- 10 and p are coprime

clearly insufficient

Combining both statements

p is square of prime number other than 2 or 5; hence Positive factors of 10 p = 12

Sufficient

Bunuel wrote:
If p is a positive integer, what is total number of positive factors of 10p

(1) p has total of 3 positive factors

(2) The greatest common factor of 10 and p is 1

Are You Up For the Challenge: 700 Level Questions
Intern
Joined: 20 Dec 2019
Posts: 35
Re: If p is a positive integer, what is total number of positive factors  [#permalink]

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27 Dec 2019, 22:26
Still unclear, howcome 8 factors ?
VP
Joined: 19 Oct 2018
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Location: India
Re: If p is a positive integer, what is total number of positive factors  [#permalink]

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27 Dec 2019, 22:34
$$N= a^x*b^y*c^z$$, where a, b and c are distinct prime numbers.
Number of factors of N= (x+1)*(y+1)*(z+1)

If $$p=2^2$$, $$10p= 2^3*5$$

Number of factors of 10p= (3+1)*(1+1)=8

Sejal16 wrote:
Still unclear, howcome 8 factors ?
Re: If p is a positive integer, what is total number of positive factors   [#permalink] 27 Dec 2019, 22:34
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