Pierādiet, ka skaitlis \(\overline{a_1a_2\cdots{}a_{3m}}\) dalās ar \(7\) (\(11\) vai \(13\)) tad un tikai tad, kad skaitlis
\[\overline{a_1a_2a_3} - \overline{a_4a_5a_6} + \cdots + (-1)^{m-1}\overline{a_{3m-2}a_{3m-1}a_{3m}}\]
dalās ar \(7\) (\(11\) vai \(13\)).Tā kā \(1000^k \equiv (-1)^k \pmod {1001}\), tad
\[\overline{a_1a_2a_3\cdots{}a_{3m-2}a_{3m-1}a_{3m}} \equiv \overline{a_1a_2a_3} \cdot 10^{3m-3} + \cdots + \overline{a_{3m-2}a_{3m-1}a_{3m}} \equiv\]
\[\equiv \overline{a_1a_2a_3} \cdot (-1)^{m-1} + \overline{a_4a_5a_6} \cdot (-1)^{m-2} + \cdots + \overline{a_{3m-2}a_{3m-1}a_{3m}} \equiv\]
\[\equiv (-1)^{m-1}\left( \overline{a_1a_2a_3} - \overline{a_4a_5a_6} + \cdots + (-1)^{m-1}\overline{a_{3m-2}a_{3m-1}a_{3m}} \right) \pmod {1001}.\]