Which of the following is a factor of 500x^3 108y^{18} 6 5x^3y^6 25x^2 15xy^6 9y^2?

To determine which of the options is a factor of the expression 500x³ 108y¹⁸ 6 5x³y⁶ 25x² 15xy⁶ 9y², we need to factor the coefficients and variables carefully.

First, let’s analyze the coefficients:

• 500 can be factored into 2² × 5³.
• 108 can be factored into 2² × 3³.
• 6 can be factored into 2 × 3.
• 5 can be expressed simply as 5.
• 25 can be factored into 5².
• 15 can be factored into 3 × 5.
• 9 can be factored into 3².

Now combining the highest powers of prime factors from all of these:

• The highest power of 2 is 2² from 500 or 108.
• The highest power of 3 is 3³ from 108.
• The highest power of 5 is 5³ from 500.

Next, for the variable parts:

• The variable x appears with the highest exponent of 3 from either 500x³ or 5x³.
• The highest exponent of y is 18 from 108y¹⁸.

Now, combining all of these factors together, the complete factorization would look like:

2² × 3³ × 5³ × x³ × y¹⁸.

After analyzing the choices, we can conclude that, if all the provided options (6, 5x³y⁶, 25x², 15xy⁶, 9y²) can be formed by these factors’ multiplications, then they are indeed factors of the overall expression.

Hence, the answer is All of the above.