Call a subset of the integers special if, for every integer $n$, every residue from $0$ to $n-1$ is obtained when all elements from the subset are taken modulo $n$. Is the set of all non-decreasing integers special? Non-decreasing integers never have digits decrease from left to right in base $10$. Examples are 11134, 44489, and 567.
It seems as if it would be true. Since I saw no method for a direct proof, I first looked for counterexamples. Then I thought that perhaps no such integer is congruent to 10 modulo 11. I searched hard and found no such example. I'm not sure how to proceed with either route (i.e. either proving the statement directly or disproving it by proving that 10 modulo 11 is never obtained).
Answer
Dunno about $10$ mod $11$. But it's obvious that the set of non-decreasing integers is not special: Consider $n=100$.
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