Can Two Different Base Representations of an Integer Prove It is Unique?

[sty2004]

base representation please help~

It is known that if a_sk^s+a_s-1k^s-1+...+a₀ is a representation of n to the base k, then 0<n<=k^s+1-1.

Now suppose n=a_sk^s+a_s-1k^s-1+...+a₀ and m=b_tk^t+b_t-1k^t-1+...+b₀ with a_s,b_t not equal to 0, are two different representations of n and m to base k, respectively. Without loss of generality we may assume t>=s. Without using Theorem 1-3(existence and uniqueness of such representation of an integer), prove directly that m not equal to n.

Many many thanks~~~

 

[telesyn]

We consider the difference :
D = m-n = c_tk^t + c_{t-1}k^{t-1} + ... + c_0
Where c_i = b_i - a_i ( i from 1 to t, and a_i = 0 if i &gt; s)

Due to the fact that a_sa_{s-1}...a_1 and b_tb_{t-1}...b_1 are two different representations, c_i must not equal to 0 for every i, otherwise a_i = b_i for all i.

Choose j is the largest number such that c_j is different from
If c_j &gt; 0,then:
D &gt;= k^j -(k-1)(k^{j-1} + ... 1) = k^j -(k^j-1) = 1 &gt;0
If c_j &lt; 0, similarly -D &gt; 0

We always have D different from 0

Good luck.