If $p$ is a positive integer and $p^2$ has total $17$ positive factors, then find the number of positive integers

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VJesus12: Legendary Member; Posts: 2276; Joined: Sat Oct 14, 2017 6:10 amFollowed by: 3 members

by VJesus12 » Sun Jul 12, 2020 12:04 am

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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

[spoiler]OA=A[/spoiler]

Source: e-GMAT

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Source: Beat The GMAT — Problem Solving | Sun Jul 12, 2020 12:04 am

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Jay@ManhattanReview: GMAT Instructor; Posts: 3008; Joined: Mon Aug 22, 2016 6:19 amFollowed by: 34 members; Location: Grand Central / New York; Thanked: 470 times

Re: If $p$ is a positive integer and $p^2$ has total $17$ positive factors, then find the number of positive integ

by Jay@ManhattanReview » Thu Jul 16, 2020 11:28 pm

VJesus12 wrote: ↑
Sun Jul 12, 2020 12:04 am
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

[spoiler]OA=A[/spoiler]

Source: e-GMAT

Note that for a number N = a^n*b^m, the no. of positive factors of N = (n + 1)(m + 1)

Since 17 is a prime number, it can only be written as (16 + 1). Thus, p^2 = a^16; where a is a prime number

=> p = a^8 => No. of factors of p = (8 + 1) = 9;

=> p^3 = a^24 => No. of factors of p^3 = (24 + 1) = 25

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

Correct answer: A

Hope this helps!

-Jay
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Scott@TargetTestPrep: GMAT Instructor; Posts: 8156; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

Re: If $p$ is a positive integer and $p^2$ has total $17$ positive factors, then find the number of positive integ

by Scott@TargetTestPrep » Sun Jul 19, 2020 11:04 am

VJesus12 wrote: ↑
Sun Jul 12, 2020 12:04 am
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

[spoiler]OA=A[/spoiler]

Solution:

We can let p = 2^8 (notice that p^2 = 2^16 has 16 + 1 = 17 factors). In this case, we see that p^3 = 2^24 and if q = 2^n where n is an integer between 9 and 24, inclusive, q divides p^3 but not p. Since there are 24 - 9 + 1 = 16 values for n, there are 16 values for q.

Answer: A

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