Simplification

The divisors of 400 are given by the prime factorization: The number of divisors of 400 is given by . The divisors are: 1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, 400. Similarly, the divisors of 1000 are: The number of divisors of 1000 is . The divisors are: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000. Now, to find the divisors common to both 400 and 1000, we calculate the greatest common divisor (gcd) of 400 and 1000. The divisors of 200 (the gcd) are the common divisors of 400 and 1000. The divisors of 200 are: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200. Thus, the number of common divisors, , is 12.

We are given that is the arithmetic mean and and are the geometric means between two positive numbers. Let's express these means as follows: Step 1: Define the means - The arithmetic mean between two numbers and is given by: - The geometric means and between the same two numbers are the numbers that satisfy: Step 2: Use the properties of means We are asked to find the value of: Using the values of and , we can simplify this expression. Step 3: Simplify the expression We know that , so we have: And for the denominator: Thus: Thus, the correct answer is .