Question of the Week- 11 ( If p is positive integer and p^2 has ..)

Question

e-GMAT          Question of the Week #11

If  p  is a positive integer and  p^2  has total  17  positive factors, then find the number of positive integers that completely divides  p^3  but does not completely divide  p ?

Options

(A)  16
 (B)  17
 (C)  21
 (D)  23
 (E)  24

To access all the questions:     Question of the Week: Consolidated List

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chetan2u · Accepted Answer

Number of factors = 17=1*17=(16+1) , 
therefore  p^2=a^16  and  p=a^8 , so p will have  (8+1) =9  factors
Now  p^3=(a^8)^3=a^24 , and factors are  (24+1)=25

Thus factors that do not divide p but divide  p^3  are 2 5-9=16

A

GyMrAT · Answer

Given  p > 0  &  p^2  has 17 factors, hence  p  is of the form  n^8  & has  (8+1) = 9  factors

we get  p^2 = (n^{8})^{2} = n^{16}

&  p^3 = (n^{8})^{3} = n^{24} , which has  (24+1)  =  25  factors

Therefore # of factors of p^3 that are not factors of  p = 25 - 9 = 16

Answer A.

Thanks,
GyM

rahul16singh28 · Answer

As  p^2  has 17 factors, so  p^2 = 2^{16}  or  p = 2^8 ........ (p can be any Prime Number)
Now,  p^3 = 2^{24} .. As per the question statement, the dividing Integer should divide  2^{24}  but not  2^8 ...... So, the dividing Integer will be  2^9, 2^{10},..... 2^{24}.

Hence, A.

Hungluu92vn · Answer

Can someone help me to elaborate more why you can conclude p^2 has 17 factors so p^2 = 2^16 or a^16. Sorry i dont get this point. Thank you so much

rahul16singh28 · Answer

Total Number of Factors for  a^{n} = n+1 .. Hope this is clear.

EgmatQuantExpert · Answer

Solution

Given:  
 • p is a positive integer
•  p^2  has 17 positive factors

To find:  
 • Number of positive integers that divides  p^3 , but does not divide p

Approach and Working:   
 • The number of factors of p^2 is 17
 o 17 is a prime number, so it can only be written as 1 * 17 = (0+1) * (16+1)
o Thus,  p^2 = P_1^{16} , where  P_1  is a prime number 
• Implies,  p = P_1^8 
 o The number of factors of p will be 9 
• And,  p^3 = P_1^{24} 
 o The number of factors of  p^3  will be 25 
• All the factors of ‘p’ will also be the factors of  p^3

Therefore, the number of positive integers that divides p^3, but does not divide p = 25 – 9 = 16

Hence, the correct answer is option A.

Answer: A

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EgmatQuantExpert · Answer

Since,  p^2 = 17 , is a prime number, 
• It can only be written as 1*17 = (0+1) * (16+1)
• And, we know that the number of factors of  N = p^a * q^b * r^c ... is (a+1) * (b+1) * (c+1) ...
Thus,  p^2 = P_1^{16} , where  P_1  is a prime number

https://e-gmat.com/blogs/wp-content/uploads/2018/08/Free-Session-NP-e1533292073470.jpg

EgmatQuantExpert · Answer

There is a small correction in your solution.

The statement "p can be any Prime Number" is incorrect, because p is not a prime number,

Instead, it should have been stated as "p can have any prime number/factor in place of 2."

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Question of the Week- 11 ( If p is positive integer and p^2 has ..)

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