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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