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# digits 0-9 in a multiple of any integer

Prove that every positive integer has a multiple whose decimal representation involves all ten digits.

Note by Sarath Ch
4 years, 2 months ago

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Suppose we have $$n\in \mathbb{N}$$ with $$m$$ digits. Now consider the number, $\large{A=123456789\underbrace{000...000}_{(m+1)~\text{zeros}}}$So exactly one $$N\in \left\{A+1, A+2, ... , A+n\right\}$$ must be a multiple of $$n$$. $$\blacksquare$$

- 4 years, 2 months ago