MathDB

You are given three lists

A

B

, and

C

. List

A

contains the numbers of the form

10^k

in base

10

, with

k

any integer greater than or equal to

1

. Lists

B

and

C

contain the same numbers translated into base

2

and

5

respectively:

\begin{array}{lll} A & B & C \\ 10 & 1010 & 20 \\ 100 & 1100100 & 400 \\ 1000 & 1111101000 & 13000 \\ \vdots & \vdots & \vdots \end{array}

Prove that for every integer

n > 1

, there is exactly one number in exactly one of the lists

B

C

that has exactly

n

digits.

5