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Suppose that $n$ is a positive integer, and $b$ is an integer $\geq 2$. Show that there exist an nonnegative integer $m$, and integers $a_{0}, a_{1}, \ldots, a_{m} \in$ $\{0,1,2, \ldots, b-1\}$ such that
$n=a_{m} b^{m}+a_{m-1} b^{m-1}+\cdots+a_{0}$
and $a_{m} \neq 0$. Moreover, show that $m$ and $a_{0}, a_{1}, \ldots, a_{m}$ are uniquely determined by $n$. (We will write $\left(a_{m} a_{m-1} \cdots a_{0}\right)_{b}$ for the right-hand side in $\left.(21)\right)$.
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