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Can you show that for all positive integers that are 10 mod 11, none of them have digits that increase at any point? I ran a Python program for numbers up to the millions checking, and I couldn't find any numbers that are 10 mod 11 that also have digits that increase at any point, so I'm inclined to believe it's true.

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  • $\begingroup$ Less confusing is to say "Can you show that all positive integers that are 10 (mod 11) have non-increasing digits". $\endgroup$
    – croraf
    Dec 15 '17 at 10:08
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    $\begingroup$ @croraf have non-increasing digits is not equivalent to (and stronger than) "do not have non-decreasing digits" which is what the question asks. Consider for example that $\,9 \cdot 11 + 10 = 109\,$, and $\,109\,$ has neither non-increasing nor non-decreasing digits. $\endgroup$
    – dxiv
    Dec 16 '17 at 1:00
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Sorry, why should the 2nd rightmost digit be $0$? Another idea: if $n$ has non-decreasing digits and at least 2 digits, let $d$ be the leftmost digit. Subtract from $n$ the number whose digits are $dd00\ldots$, with enough $0$s to make it the same length as $n$. The result is congruent to $n$ modulo $11$, still has non-decreasing digits, and has fewer digits than $n$. Keep doing this and we get down to a single digit or $0$, which is obviously not congruent to $10$ modulo $11$.

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  • $\begingroup$ why should the 2nd rightmost digit be 0 If that's about my now-deleted answer, then that was wrong and I removed it. $\endgroup$
    – dxiv
    Dec 16 '17 at 0:56

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