How many positive integers less than 1000 have exactly 5 positive divi

Question

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

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

Archit3110 · Answer

( prime ) ^4 will have exactly 5 +ver divisors
prime ; 2,3,5
option A

gmatophobia · Answer

For an integer n, the number of positive divisors ,d, is given by

n = p^x

d = x + 1

p is a prime number

Its given that d = 5, so x = 4

Hence we have to find the number of primes such that

0<  p^4  < 1000

p = 2 ;  p^4  = 16
p = 3 ;  p ^ 4  = 81
p = 5 ;  p^4  = 625

Hence the number of integers = 3

IMO A

BrentGMATPrepNow · Answer

-----------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

pappal · Answer

can be reiterated as how many integers less than 1000 have only 05 factors.
since #(N)=5 only possible combination of powers for N should be P^4 where P is a prime no.
i.e. 2^4,3^4 and 5^4 are the only ones less than 1000.
A

RajivJoarder · Answer

We need to find out X where 0 < X < 1000 such that -

X = (prime)^4
 
As in that case only, the total number of factors will be equal to 5
Example
2^4 = 16  -Number of factors = (5) 
3^4 = 81  -Number of factors = (5) 
5^4 = 625  -Number of factors = (5) 
But
 4^4 = 256  -Number of factors = (9)

Also, if we go further up to the fourth power of the next Prime number, which is 7, its greater than 1000

Therefore, 
 A is the correct answer

How many positive integers less than 1000 have exactly 5 positive divi

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How many positive integers less than 1000 have exactly 5 positive divi

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