Proof about powers of 10-s

Revision en7, by snorkel, 2021-04-10 08:11:59

Recently I was solving the problem which said that I have to find the smallest number consisting of only 1-s (1, 11, 111, ...) which will be divisible by the given number $$$n$$$. This problem is easy, but I was surprised that it is possible for any $$$n$$$ not divisible $$$2$$$ and $$$5$$$. (at least for $$$<= 100 000$$$). How can I prove this?

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  Rev. Lang. By When Δ Comment
en7 English snorkel 2021-04-10 08:11:59 1 Tiny change: 'number $n$ This prob' -> 'number $n$. This prob'
en6 English snorkel 2021-04-10 08:11:35 54
en5 English snorkel 2021-04-10 08:11:07 26 Tiny change: 'number $n$. This pr' -> 'number $n$ not divisible $2$ and $5$. This pr'
en4 English snorkel 2021-04-10 07:54:51 56
en3 English snorkel 2021-04-10 07:54:04 39
en2 English snorkel 2021-04-10 07:53:16 28 Tiny change: 'number $n$. This pro' -> 'number $n$ (at least for ${le} 100 000$). This pro'
en1 English snorkel 2021-04-10 07:50:17 323 Initial revision (published)