The question asks for the probability that a randomly chosen positive divisor of $10^{23}$ is an integral multiple of $10^{16}$.

Step-by-step solution:

1. Prime Factorization of Powers of 10: $10^{23} = 2^{23} \times 5^{23}$ $10^{16} = 2^{16} \times 5^{16}$ So, the divisors of $10^{23}$ are of the form $2^a \times 5^b$, where $0 \leq a \leq 23$ and $0 \leq b \leq 23$.

2. Number of Divisors of $10^{23}$: The total number of divisors of $10^{23}$ can be found by using the formula for the number of divisors: $\text{Number of divisors of } 10^{23} = (23+1)(23+1) = 24 \times 24 = 576$ So, there are 576 divisors of $10^{23}$.

3. Condition for Being a Multiple of $10^{16}$: For a divisor of $10^{23}$ to be a multiple of $10^{16}$, the divisor must be of the form $2^a \times 5^b$, where: $a \geq 16 \quad \text{and} \quad b \geq 16$ This is because any multiple of $10^{16}$ must contain at least $2^{16} \times 5^{16}$.

4. Number of Divisors That Are Multiples of $10^{16}$: The values of $a$ and $b$ can range from 16 to 23, inclusive. So, both $a$ and $b$ have 8 possible values (16, 17, ..., 23).

   Therefore, the number of divisors that are multiples of $10^{16}$ is: $8 \times 8 = 64$

5. Probability Calculation: The probability is the ratio of favorable outcomes (divisors that are multiples of $10^{16}$) to the total outcomes (all divisors of $10^{23}$): $\text{Probability} = \frac{64}{576} = \frac{1}{9}$

Thus, the probability that a randomly chosen positive divisor of $10^{23}$ is an integral multiple of $10^{16}$ is $\frac{1}{9}$.

Answer: $\boxed{\frac{1}{9}}$

Would you like further clarification on any of these steps?

Here are 5 related questions for deeper understanding:

1. How do you find the number of divisors of a number?
2. What is the significance of prime factorization in solving this type of problem?
3. How would the probability change if the powers in the problem were different, say $10^{30}$ and $10^{20}$?
4. How do you apply combinatorics to divisor-related probability problems?
5. What is the formula for the number of divisors of a number given its prime factorization?

Tip:

When solving divisor-related probability questions, always break the problem into prime factorizations and count the ranges of exponents carefully.