Question

The number of numbers below 1000 which have exactly 5 factors

Answer 1

The no. which have a perfect root can have 5 factors and moreover, their square root should have 3 factors.

Answer 2

Heya User,

--> Suppose, if you're thinking that :->

--> 630 = 2*5*7*9  --> is a number having exactly five factor.. 
  --> *_* I'm afraid you're wrong....

After all -->  63,10,14,etc. are also the factors... 

So, we have the following result... --> 

$-\ \textgreater \ If \ n = p_1^{ \alpha _1}p_2^{ \alpha _2}...p_n^{ \alpha _n}$, where p₁p₂p₃p₄...p(n) are all primes.. 
Then, no. of positive factors of 'n' = (α₁+1)(α₂+1)(α₃+1)...(α(n)+1)

However, if a number has five factors ,
 --> 5 = 1 * 4  --> and hence, we get an idea that .... +_+

---> The required numbers are perfect fourth powers...

=> All such numbers with exactly five factors, below 1000 are :->
         --> 1⁴ , 2⁴, 3⁴, 4⁴, 5⁴

Hence, we see that =_= there are only five such numbers below 1000 which have exactly five factors