The probability that a randomly chosen positive divisor of $$10^{2023}$$ is an integer multiple of $$10^{2001}$$ is

$$10^{2023}=2^{2023}\cdot5^{2023}$$

$$10^{2001}=2^{2001}\cdot5^{2001}$$

$$10^{2023}$$ can be written as $$10^{2001}*2^{22}\cdot5^{22}$$

The number of factors of $$10^{2023}$$ that are multiple of $$10^{2001}$$ will be the number of factors of  $$2^{22}\cdot5^{22}$$

Number of factors of $$2^{a}\cdot5^{b} = (a+1)(b+1)$$

Number of factors of  $$2^{22}\cdot5^{22}$$ = 23*23 = 529

And the total number of factors for $$10^{2023}=2^{2023}\cdot5^{2023}$$ is (2023+1)*(2023+1) = $$2024^2$$

Probability = $$\frac{529}{2024^2}$$