Positive integers are the counting numbers, the numbers we use to
count with: 1,2,3,4,5,6, etc. (ad infinitum)

1 is a factor of every positive integer, and the square of 1 is 1,
and so 1 fits the requirements.  So one such integer is

1. 1

To find the others,

we break 2000 down into prime factors.

We get 24×53

The perfect squares we can make of those are 

2.  22, 
3.  24 = (22)2 = 42 
4.  52

And when we multiply two squares, we get another square, so we can
also get:
  
5. 2252 = (2×5)2 = 102,
6. 4252 = (4×5)2 = 202,

So there are 6 such integers:

1.  1 is such because 12 = 1 = 2000÷2000
2.  2 is such because 22 = 4 = 2000÷500
3.  4 is such because 42 = 16 = 2000÷125
4.  5 is such because 52 = 25 = 2000÷80
5.  10 is such because 102 = 100 = 2000÷20
6.  20 is such because 202 = 400 = 2000÷5

Edwin