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Problem 714

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Duodigits

We call a natural number a duodigit if its decimal representation uses no more than two different digits. For example, $12$, $110$ and $33333$ are duodigits, while $102$ is not.
It can be shown that every natural number has duodigit multiples. Let $d(n)$ be the smallest (positive) multiple of the number $n$ that happens to be a duodigit. For example, $d(12)=12$, $d(102)=1122$, $d(103)=515$, $d(290)=11011010$ and $d(317)=211122$.

Let $\displaystyle D(k)=\sum_{n=1}^k d(n)$. You are given $D(110)=11\ 047$, $D(150)=53\ 312$ and $D(500)=29\ 570\ 988$.

Find $D(50\ 000)$. Give your answer in scientific notation rounded to $13$ significant digits ($12$ after the decimal point). If, for example, we had asked for $D(500)$ instead, the answer format would have been $2.957098800000e7$.

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双字数

如果一个自然数的十进制表示使用了不超过两种不同的数字，则称这个数为双字数。例如，$12$、$110$和$33333$都是双字数，而$102$则不是。
可以证明，任何自然数都存在一个为双字数的倍数。记$d(n)$为$n$的正倍数中最小的双字数。例如，$d(12)=12$，$d(102)=1122$，$d(103)=515$，$d(290)=11011010$，$d(317)=211122$。

记$\displaystyle D(k)=\sum_{n=1}^k d(n)$。已知$D(110)=11\ 047$，$D(150)=53\ 312$，$D(500)=29\ 570\ 988$。

求$D(50\ 000)$，并将你的答案用科学计数法表示，保留$13$位有效数字（也即小数点后$12$位）。例如，若我们要求的答案是$D(500)$，则答案应当写作$2.957098800000e7$。

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