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