Solution:
Any number N has 1 and N as 2 of its factors.
Let x be the third factor.
So N/x should also be a factor of N.
But since N has only 3 distinct factors, N/x can be 1, x or N.
If N/x = 1, x = N. But N has been considered as a factor. So eliminate this case.
If N/x = N, x = 1. Again we have taken 1 as factor. So rule out this also.
The only possibility is N/x = x or N = x^2. So N has to be a perfect square.
Note that x will be a prime number.
Else one of its factors will become a fourth factor of N.
So N is square of a prime number.
Now a perfect square < 1000 is 961.
Or N <= 961.
961 = 31 ^ 2.
So the primes <= 31 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31.
The squares of the above 11 numbers between 1 - 1000 will have exactly 3 different factors.