Prove that any base $a>1$ , each positive integer $m$ has a unique expression of the form $a^nr_n+a^{n-1}r_{n-1}+...+ar_1+r_0$

Question

I have the next problem. Prove that any base $a>1$, each positive integer $m$ has a unique expression as the form $$a^nr_n+a^{n-1}r_{n-1}+...+ar_1+r_0$$ where the integers $r_k$ satisfy $$0\le r_k \lt a, r_n\neq0.$$ I have tried this.

Proof: By induction above n $$m_1=ar_1+r_0$$

We suppose that it is true for n=k

$$m_k=a^kr_k+a^{k-1}r_{k-1}+...+ar_1+r_0$$ Prove for n=k+1. By hypothesis $$m_k=a^kr_k+a^{k-1}r_{k-1}+...+ar_1+r_0$$ We add $a^{k+1}r_{k+1}$.

$$m_k+a^{k+1}r_{k+1}=a^{k+1}r_{k+1}+a^kr_k+a^{k-1}r_{k-1}+...+ar_1+r_0$$

As $a^{k+1}r_{k+1}$ and $m_k$ are positive integers then be $m_{k+1}=m_k+a^{k+1}r_{k+1}$ a positive integer, hence

$$m_{k+1}=a^{k+1}r_{k+1}+a^kr_k+a^{k-1}r_{k-1}+...+ar_1+r_0$$

but I am not sure and still I need to prove the uniqueness.Thank you for your advices.

Answer 1

In short, the coefficients $r_i$ are the remainders of the repeated division of $m$ by $a$. Then what you says follows from the uniqueness of the remainder in the Euclidean division.