Problem 2

Let

a,b,n>1

be integers and let

a,b

be bases of two number systems. Using the same digits

x_n,\ldots,x_0

, with

x_n\ne0

and

x_{n-1}\ne0

, define

A_n=x_nx_{n-1}\cdots x_0

and

A_{n-1}=x_{n-1}\cdots x_0

in base

a

, and

B_n=x_nx_{n-1}\cdots x_0

and

B_{n-1}=x_{n-1}\cdots x_0

in base

b

. Prove that

\frac{A_{n-1}}{A_n}<\frac{B_{n-1}}{B_n}

if and only if

a>b

Step 5 of 5: Finish the iff statement

In plain words

The three possible orders of the bases exhaust all cases.

a=b\\Rightarrow\\frac{A_{n-1}}{A_n}=\\frac{B_{n-1}}{B_n};\\quad a<b\\Rightarrow\\frac{A_{n-1}}{A_n}>\\frac{B_{n-1}}{B_n}

Detailed analysis

When $a=b$ , the representations are identical. When $a<b$ , exchange the roles of $a$ and $b$ in the proved direction, reversing the inequality.