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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 = $$\frac{529}{2024^2}$$
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