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For some arbitrary natural numbers $n,i$ with $n<k,i \leq k+1$, does there always exists some natural $x \leq k-1$ such that $k \mid n+xi$?

So we want $ xi \equiv -n \pmod k$. Reduced residue class of $k$ consist ${0,1,2,...,k-1}$, that is $k$ elements, so with all the bounds given in the problem, I was thinking of getting some PHP argument, but I'm stuck. Any help would be appreciated.

edited Oct 06 '23 at 14:43

asked Oct 06 '23 at 13:17

zaemon_23

Note that there are $k-1$ elements that you listed out. Maybe you forgot to include 0? – Calvin Lin Oct 06 '23 at 14:12
Hint: No. $\quad$ Now go find a counter example. How can we ensure that $k \not \mid n+xi$? – Calvin Lin Oct 06 '23 at 14:14
@CalvinLin oh yeah you're right, i did forget to include 0. – zaemon_23 Oct 06 '23 at 14:43
See here in the linked dupe for necessary (and sufficient) conditions for solvability of a linear congruence. – Bill Dubuque Oct 06 '23 at 17:37

For some arbitrary natural numbers $n,i$ with $n

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