Question

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 \(\left.(21)\right)\).