How many three digit numbers have exactly three factors?

Every number has two factors: 1 and the number itself. Therefore, for a 3-digit number with 3 factors we need to find exactly one factor other than these two factors. Now, factors of a non-square number occur in pairs. Thus, if x is a factor of n, then n/x is also a factor of n.
Since the required number has only one factor other than 1 and the number itself, therefore x and n/x are equal, otherwise the number will have 4 factors: 1, the number itself(n), x, n/x, where x is a factor. Therefore, the 3-digit number is not a non-square number.

Also, since the number has no other factor, x has to be a prime number.

Now, the perfect squares with only one prime factor are the squares of the prime numbers.

The prime numbers with 3-digit squares are 11, 13, 17, 19, 23, 29 and 31. The squares of these prime numbers are 121, 169, 289, 361, 529, 841 and 961 respectively. Clearly, these all perfect square have only three factors: 1, the perfect square itself and the prime number whose square makes the perfect square.

Therefore, there are seven 3- digit numbers 121, 169, 289, 361, 529, 841 and 961 that have only 3-factors.