On the solvability of the equation $\sum ^ n_ {i=1}x_ i/d_ i\equiv 0\;(\textrm {mod} 1)$ and its application

Abstract:

In this paper, we obtain a necessary and sufficient condition under which the equation of the title is unsolvable. More precisely, for the equation \[ \frac {{{x_1}}}{{{d_1}}} + \frac {{{x_2}}}{{{d_2}}} + \cdots + \frac {{{x_n}}}{{{d_n}}} \equiv 0\quad (\bmod 1),\quad {x_i}{\text { integral}},{\text {1}} \leq {x_i} < {d_i}(1 \leq i \leq n),\] where ${d_1}, \ldots ,{d_n}$ are fixed positive integers, we prove the following result: The above equation is unsolvable if and only if 1. For some ${d_i},({d_i},{d_1}{d_2} \cdots {d_n}/{d_i}) = 1$, or 2. If ${d_{{i_1}}}, \ldots ,{d_{{i_k}}}(1 \leq i < \cdots < {i_k} \leq n)$ is the set of all even integers among $\left \{ {{d_1}, \ldots ,{d_n}} \right \}$, then $2\nmid k,{d_{{i_1}}}/2, \ldots ,{d_{{i_k}}}/2$ are pairwise prime, and ${d_{{i_j}}}$ is prime to any odd number in $\{ {d_1}, \ldots ,{d_n}\} (j = 1, \ldots ,k)$.