MathDB

Let

({{A}_{n}})_{n=1}^{\infty }

and

({{a}_{n}})_{n=1}^{\infty }

be sequences of positive integers. Assume that for each positive integer

x

, there is a unique positive integer

N

and a unique

N-tuple

({{x}_{1}},...,{{x}_{N}})

such that

0\le {{x}_{k}}\le {{a}_{k}}

for

k=1,2,...N

{{x}_{N}}\ne 0

, and

x=\sum\limits_{k=1}^{N}{{{A}_{k}}{{x}_{k}}}

.
(a) Prove that

{{A}_{k}}=1

for some

k

; (b) Prove that

{{A}_{k}}={{A}_{j}}\Leftrightarrow k=j

; (c) Prove that if

{{A}_{k}}\le {{A}_{j}}

, then

\left. {{A}_{k}} \right|{{A}_{j}}

1