#### Vol. 3, No. 3, 2010

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Distinct solution to a linear congruence

Vol. 3 (2010), No. 3, 341–344
##### Abstract

Given $n,k\in ℕ$ and ${a}_{1},{a}_{2},\dots ,{a}_{k}\in {ℤ}_{n}$, we give conditions for the equation ${a}_{1}{x}_{1}+{a}_{2}{x}_{2}+\cdots +{a}_{k}{x}_{k}=1$ in ${ℤ}_{n}$ to admit solutions with all the ${x}_{i}$ distinct.

A sufficient condition is that $k\le \varphi \left(n\right)$ and ${a}_{i}$ be invertible in ${ℤ}_{n}$ for all $i$.

If $n>2$ is prime, the following conditions together are necessary and sufficient: $k\le n$, each ${a}_{i}$ is nonzero, and either $k or not all of the ${a}_{i}$ are equal.

##### Keywords
linear congruence, minimal zero-sum sequence, property B
##### Mathematical Subject Classification 2000
Primary: 11B50, 11D79