Brent@GMATPrepNow wrote: ↑

Sun Oct 30, 2022 6:16 am

How many positive integers less than 1000 have exactly 5 positive divisors?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

Answer: A
Source: www.gmatprepnow.com

-----------ASIDE------------------
If the prime factorization of N = (p^a)(q^b)(r^c) . . . (where p, q, r, etc are different prime numbers), then N has a total of (a+1)(b+1)(c+1)(etc) positive divisors.

Example: 14000 = (2^4)(5^3)(7^1)
So, the number of positive divisors of 14000 = (4+1)(3+1)(1+1) =(5)(4)(2) = 40
---------------------------------

If a number has exactly 5 positive divisors, then (a+1)(b+1)(c+1)(etc) = 5
Important: There is only ONE way to factor 5. That is 5 = (5)(1)
So, the ONLY way to write 5 as (a+1)(b+1)(c+1)(etc) is as follows...
5 = (4+1)(0+1)(0+1)(etc)
In other words, if N = a prime number raised to the power of 4, then N will have (4+1) positive divisors (i.e., 5 positive divisors)

For example, if N = 2^4 (aka 16) then N has 5 divisors: 1, 2, 4, 8 and 16
Likewise, if N = 3^4 (aka 81) then N has 5 divisors: 1, 3, 9, 27 and 81
Likewise, if N = 5^4 (aka 625) then N has 5 divisors: 1, 5, 25, 125, and 625
Likewise, if N = 7^4 (aka some number greater than 1000) then N has 5 divisors, BUT we can't include this value of N, since its greater than 1000.

So, there are three numbers (16, 81, and 625) less than 1000 that have exactly 5 positive divisors.

Answer: A