We are tasked with finding the probability that a randomly chosen positive divisor of $10^{23}$ is an integral multiple of $10^{16}$.

Step 1: Prime Factorization of $10^{23}$

The number $10^{23}$ can be written as: $10^{23} = (2 \cdot 5)^{23} = 2^{23} \cdot 5^{23}$ So, the divisors of $10^{23}$ are of the form $2^a \cdot 5^b$, where $0 \leq a \leq 23$ and $0 \leq b \leq 23$.

Step 2: Total Number of Divisors of $10^{23}$

The total number of divisors of $10^{23}$ is given by the formula for the number of divisors of a number $n = p^a \cdot q^b$, which is $(a+1)(b+1)$. In this case, the total number of divisors is: $(23+1)(23+1) = 24 \times 24 = 576$

Step 3: Condition for Divisors to be Multiples of $10^{16}$

A divisor of $10^{23}$ is an integral multiple of $10^{16} = 2^{16} \cdot 5^{16}$. Thus, we need the divisor to be of the form $2^a \cdot 5^b$ where $a \geq 16$ and $b \geq 16$.

• For $a$, the possible values are $a = 16, 17, \dots, 23$, which gives $23 - 16 + 1 = 8$ options.
• Similarly, for $b$, the possible values are $b = 16, 17, \dots, 23$, which also gives $8$ options.

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

Step 4: Probability Calculation

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

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

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Do you want more details or have any questions?

Here are 5 related questions to explore:

1. What is the prime factorization of $10^{16}$?
2. How do you calculate the number of divisors of a number?
3. What is the probability that a randomly chosen divisor of $10^{23}$ is an integral multiple of $2^5$?
4. How does the probability change if we consider $10^{25}$ instead of $10^{23}$?
5. Can the method used here be applied to other prime factorizations?

Tip: To calculate the number of divisors of a number, express it in its prime factorized form and use the divisor formula.