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.

1; 35 is the ratio

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