This is from the book: Elementary number theory, David M. Burton, page 69.
Given an integer $b>1$, any positive integer $N$ can be written uniquely in terms of powers of $b$ as $N=a_{m}b^{m}+a_{m-1}b^{m-1}+\cdots +a_{2}b^{2}+a_{1}b+a_{0},$ where the coefficients $a_{k}$ can take on the $b$ different values $0,1,2,\dots,b-1$.
For the Division Algorithm yields integers $q_{1}$ e $a_{0}$ satisfying $N=q_{1}b+a_{0},\,0\leq a_{0}< b.$ If $q_{1}\geq b,$ we can divide once more, obtaining $q_{1}=q_{2}b+a_{1},\, 0\leq a_{1}< b.$ Now substitute >for $q_{1}$ in the earlier equation to get $N=(q_{2}b+a_{1})b+a_{0}=q_{2}b^{2}+a_{1}b+a_{0}$ As long as $q_{2}\geq b,$ we can continue in the same fashion. Going one more step: $q_{2}=q_{3}b+a_{2}$, where $0\leq a_{2}; hence $N=q_{3}b^{3}+a_{2}b^{2}+a_{1}b+a_{0}$ Because $N>q_{1}>q_{2}>\cdots \geq 0$ is a strictly decreasing sequence of integers, this process must eventually terminate, say, at the $(m-1)$th stage, where $q_{m-1}=q_{m}b+a_{m-1},\,0\leq a_{m-1} and $0\leq q_{m}. Setting $a_{m}=q_{m}$, we reach the representation $N=a_{m}b^{m}+a_{m-1}b^{m-1}+\cdots +a_{2}b^{2}+a_{1}b+a_{0}$ which was our aim.
After that the author proves the uniqueness.
My question is, once the integers $a_{i}$ and $q_{i}$ are unique, is this not enough to ensure the uniqueness of the representation?