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?