Question 752568
<pre>
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 2<sup>4</sup>×5<sup>3</sup>

The perfect squares we can make of those are 

2.  2<sup>2</sup>, 
3.  2<sup>4</sup> = (2<sup>2</sup>)<sup>2</sup> = 4<sup>2</sup> 
4.  5<sup>2</sup>

And when we multiply two squares, we get another square, so we can
also get:
  
5. 2<sup>2</sup>5<sup>2</sup> = (2×5)<sup>2</sup> = 10<sup>2</sup>,
6. 4<sup>2</sup>5<sup>2</sup> = (4×5)<sup>2</sup> = 20<sup>2</sup>,

So there are 6 such integers:

1.  1 is such because 1<sup>2</sup> = 1 = 2000÷2000
2.  2 is such because 2<sup>2</sup> = 4 = 2000÷500
3.  4 is such because 4<sup>2</sup> = 16 = 2000÷125
4.  5 is such because 5<sup>2</sup> = 25 = 2000÷80
5.  10 is such because 10<sup>2</sup> = 100 = 2000÷20
6.  20 is such because 20<sup>2</sup> = 400 = 2000÷5

Edwin</pre>