Consider all natural numbers whose decimal ex | vedclass

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(B) Let the set of numbers be $S = \{2, 4, 6, 8, 20, 22, 24, 26, 28, 40, \dots \}$.These numbers are formed using $5$ digits $\{0, 2, 4, 6, 8\}$.Since

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$\log_5 10$

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(B) Let the set of numbers be $S = \{2, 4, 6, 8, 20, 22, 24, 26, 28, 40, \dots \}$.These numbers are formed using $5$ digits $\{0, 2, 4, 6, 8\}$.Since $0$ cannot be the leading digit,the $n$-th number $a_n$ can be related to the base-$5$ representation.Let $n$ be represented in base $5$ as $(d_k d_{k-1} \dots d_0)_5$. The $n$-th number $a_n$ is obtained by mapping the digits $0, 1, 2, 3, 4$ to $0, 2, 4, 6, 8$ respectively.For large $n$,$a_n \approx 2 \cdot 10^{\log_5 n}$.Taking the logarithm,$\log a_n \approx \log 2 + \log_5 n \cdot \log 10 = \log 2 + \frac{\log n}{\log 5} \cdot \log 10$.Dividing by $\log n$ and taking the limit as $n \rightarrow \infty$:$\lim_{n \rightarrow \infty} \frac{\log a_n}{\log n} = \frac{\log 10}{\log 5} = \log_5 10$.

Consider all natural numbers whose decimal expansion has only the even digits $0, 2, 4, 6, 8$. Suppose these are arranged in increasing order. If $a_n$ denotes the $n$-th number in this sequence,then $\frac{\lim_{n \rightarrow \infty} \log a_n}{\log n}$ equals

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