Proving the Uniqueness of Base Representation for Integers

[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~~~

 

[dodo]

You can try dividing both numbers by k^s+1... one quotient will be <1 and the other >=1.

Edit: Oh, I'm sorry. I thought t>s, while you said instead t>=s. In this case I think you need something else, because if t=s, a_s=b_s and all a_i,b_i are 0 for i<s, then n=m.

Oh, but then again... you said they are different representations. I assume this means that, even if t=s, a_s and b_s cannot be the same. Then, in this case, divide both numbers by max(a_s,b_s) . k^s, and that should give you one quotient <1 and another >=1.