There are 1,963 such integers.
Every factor of a number has a pair. The only time there will be an odd number of factors is if one factor is repeated, ie the number is a perfect square.
So the question is really asking: how many positive integers less than 2008 (in the range 1 to 2007) are not perfect squares.
√2007 = 44 and a bit (it lies between 44 and 45)
So there are 44 integers less than (or equal to) 2007 which are perfect squares
2007 - 44 = 1963 integers are not perfect squares in the range 1-2007 and have an even number of factors (divisors).