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

Question

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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Archit3110 · Answer

#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 
or it can be 2*2 * 3 ; 12 factors 
insufficient
#2
The greatest common factor of 10 and p is 1
insufficient

from 1 &2
total factors would be 10*p ; 2*5* p ; 2*2*3; 12 factors
OPTION C

Sejal16 · Answer

Here the question is not about different positive factors, its asking only positive factors - ie they might repeat.
Hence the answer should be A

nick1816 · Answer

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

Sejal16 · Answer

Still unclear, howcome 8 factors ?

nick1816 · Answer

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

Anurag06 · Answer

Hi

How did you assume that if p = 2^2 when it is not mentioned?

Can you please clarify.

Thanks.

Aryaa03 · Answer

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

Number of total factors for a number in its prime factorization form  a^p.b^q.c^r  is given as n=(p+1)(q+1)(r+1)
Prime factors of 10p = prime factors of 10 ( 2^1.5^1)  AND prime factors of p. 
 1) St-1: 
 p has a total of 3 positive factors : If a number has odd number of distinct factors then it's a perfect square and if the number of factors is 3, it can be further inferred that it is prime square
 Now following case arise:
  Case:1 
 p is 4 or 25 i.e. has a prime factorization of  2^2*5  or  2*5^2  . Accordingly,10p will have prime factorization as:  2^3*5^1  or  2^1*5^3  and n=8 
  Case-2: 
 p is perfect square of any other prime i.e. has factors ( 1,a, a^2) , then 10p will have prime factorization as:  2^1*5^1*a^2  and n=12
 n can be 8 or 12, this statement is  INSUFFICIENT 
 2) St-2: 
 Greatest common factor of 10 and p are 1.
 p can be any number other than a multiple of 2 or 5. Therefore it can be for example 3, where no. of total factors will be 8 or p can be 9 ( 3^2 ), where number of total factors will be 12.
 This is clearly  INSUFFICIENT .
 
Combining St-1 & 2, 
 From (2), it is known that p is neither a multiple of 2 nor 5, so case 2 of statement-1 given n=12.  SUFFICIENT  
 .

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

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GMAT Data Sufficiency (DS) Questions

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

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