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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
2 years, 12 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 \(n\). \(\blacksquare\) Jubayer Nirjhor · 2 years, 12 months ago

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