2008 APMO 第 5 题

Let $a, b, c$ be integers satisfying $0<a<c-1$ and $1<b<c$. For each $k$, $0 \leq k \leq a$, let $r_{k}, 0 \leq r_{k}<c$, be the remainder of $k b$ when divided by $c$. Prove that the two sets $\left\{r_{0}, r_{1}, r_{2}, \ldots, r_{a}\right\}$ and $\{0,1,2, \ldots, a\}$ are different.

令$a、b、c$为满足$0<a<c-1$和$1<b<c$的整数。对于每个 $k$, $0 \leq k \leq a$，令 $r_{k}, 0 \leq r_{k}<c$ 为 $k b$ 除以 $c$ 时的余数。证明两个集合 $\left\{r_{0}, r_{1}, r_{2}, \ldots, r_{a}\right\}$ 和 $\{0,1,2, \ldots, a\}$ 不同。