Only squares of primes have 3 factors, 

The three factors of the square of a prime are 

1. 1
2. the prime 
3. the number itself, which is the square of that prime.

Prime numbers have only 2 factors and composite numbers have at least 4
factors. (For example 6 has 4 factors 1,2,3,6) 

The square root of 2010 is about 44.8...

So we list all the primes less than 44.8 and square them.

Those primes are

2,3,5,7,11,13,17,19,23,29,31,37,41,43.

So the answer is 14, because when we square them like this:

4,9,25,49,121,169, 289,361,529,841,961,1369,1681,1849.

we have 14 squares of primes less than 2010, all of which have exactly
3 factors.

Answer: 14

Edwin