If p is a positive integer and p^2 has total 17 positive factors, then

Question

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?

A. 16
B. 17
C. 21
D. 23
E. 24

chetan2u · Answer

Number of factors =17 
It means we are looking at some square and it is given as  p^2 .
17 can be broken as multiple of different numbers as only 1*17..

Conversion of number of factors to the exponent 
17=1*17=(16+1)
So,  p^2=a^{16}.....p=a^8 , where a is a prime number.

Value of  p^3=(a^8)^3=a^{24} ..
All values between  a^9  and  a^{24} , both inclusive will divide  a^{24}  but not  a^8 ..
so  a^9, a^{10},a^{11}...a^{23},a^{24} 
Number of values = 24-9+1=16

A

Kritisood · Answer

Number of factors of p^2 is 17
17 can only be written as 17*1
hence p^2 must be equal to P^16 as 16+1 = 17 (where P is a prime no.)
then p = P^8 (no of factors = 8+1=9)
and p^3 = 24 (no of factors = 24+1=25)
Therefore, the no of positive integers that divides p^3, but does not divide p = 25 – 9 = 16

Kinshook · Answer

Asked: 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?

p^2 = a^16
p = a^8

Positive integers that completely divides (p^3=a^24) but does not completely divide (p=a^8) = {a^9, a^10,.... a^24}
Number of positive integers that completely divides (p^3=a^24) but does not completely divide (p=a^8) = 24-9+1 = 16

IMO A

jainpearl · Answer

hi, can you explain why you did P^8 and the steps after that?

Bunuel · Answer

The use of capital "P" in that solution is confusing.

Consider this:

p^2 having 17 total positive factors implies p^2 = n^16, for some prime number n.

Taking the square root gives p = n^8, which means that p has 9 positive factors.

p^3 would equal n^24, and thus would have 25 positive factors.

The difference in the number of factors is therefore 25 - 9 = 16.

Answer: A.

Akshaynandurkar · Answer

Very good question.
we have 17 positive factors for p^2, 
prime factored form for p^2 = ( some Prime no )^16 (since we total factors = 16+1 = 17)
that means p^2 = x^16
therefore p = x^8
and hence p^3 = x^24
total number of factors for p and p^3 are 9 and 25 respectively 
since 9 factors of p are common to p^3 factors, they should not be included in the list 
so total number of factors that divide p^3 and not p = 25-9 = 16

bumpbot · Answer

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If p is a positive integer and p^2 has total 17 positive factors, then

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If p is a positive integer and p^2 has total 17 positive factors, then

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