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Prove that for each $n \in \mathbb{N}, s \in \mathbb{N}$ the following is true

(i) $n \equiv Q_s(n) \left(\bmod\ 10^s - 1\right)$

(ii) $n \equiv Q'_s(n)\left(\bmod\ 10^s + 1\right)$

where

$$Q_s(n) = \sum_{i=0}^{\infty}(a_{is+s-1}\dots a_{is+1}a_{is})$$

for example

$$Q_3 (6154328103) = 103 + 328 + 154 + 006 = 591$$

and

$$Q'_s(n) = \sum_{i=0}^{\infty}(-1^i)(a_{is+s-1}\dots a_{is+1}a_{is})$$

for example

$$Q'_3 (6154328103) = 103 - 328 + 154 - 006 = -77$$

Also $n$ can be expressed as

$$n = \sum_{i=0}^{\infty}(a_{is+s-1}\dots a_{is+1}a_{is}) \cdot 10^{is}$$

for example

$$6154328103 = 103 \cdot 10^{0 \cdot 3} + 328 \cdot 10^{1 \cdot 3} + 154 \cdot 10^{2 \cdot 3} + 6 \cdot 10^{3 \cdot 3}$$

Any ideas?

  • 2
    Do you recognize these as generalizations of the usual tests for divisibility by $9$ and by $11$? I expect you have seen the proofs of correctness for those.2012-02-28
  • 0
    A biologist, eh? Anyways, you should at least explain that the $a_k$'s stand for the digits in base expansions of the argument.2012-02-28

4 Answers 4