A generalization

Prove that any positive integer x lying between a^k and a^k+1 can be represented uniquely as follows

x = a^k + b_k-1 a^k-1 +...+b_k-r a^k-r +..+b₀ a<sup>0

where a is a positive integer >1 and each b_i (non negative integer) is less than a.

I generalized this based on the problems found in Niven's number theory book( pg 19 pro, 44 and 45 , fifth edition)</sup>