There are $n$'s numbers among $\left\{ {1,10,{{10}^2},{{10}^3},...} \right\}$ congruent modulo $n$; we can take $m$ to be the sum of them.

Can anyone explain the above solution more elaborated

any ideas Mathlinkers

yunxiu wrote: There are $n$'s numbers among $\left\{ {1,10,{{10}^2},{{10}^3},...} \right\}$ congruent modulo $n$; we can take $m$ to be the sum of them. i didn't really get what you said..........doesn't make much sense to me..........maybe im wrong

You can always find such $x_1<x_2<...<x_n$ that $10^{x_1} \equiv 10^{x_2} \equiv ... \equiv 10^{x_n}=t \pmod {n}$ for some $t$ So you can build $m=10^{x_1}+...+10^{x_n}$ $m$ consists of $n$ ones and some zeroes and $m \equiv nt \equiv 0 \pmod n$