What fraction of the positive integer factors of 1000^3, are perfect squares?

25/81

Step-by-step explanation:

1000³ = (10³) ³ = 10⁹ = 2⁹ * 5⁹

For a factor to be a perfect square, the exponents must be even. For example:

2⁰ * 5⁰, 2⁰ * 5², 2² * 5⁰, etc.

There are 5 even exponents between 0 and 9 for each 2 and 5. So the number of perfect square factors is 5*5 or 25.

The total number of factors is 9*9 or 81.

So the fraction of positive integer factors that are perfect squares is 25/81.