### Uzdevums:

6p.
Kādu atlikumu dod skaitlis ${3}^{61}$ dalot ar $$7$$?

Risini pakāpeniski!
1) Atrodi virknes ${3}^{n}$ periodu pēc moduļa $$7$$.

Ja $$n=0$$, tad  ${3}^{0}\equiv i\phantom{\rule{0.147em}{0ex}}\left(\mathit{mod}\phantom{\rule{0.147em}{0ex}}7\right)$
Ja $$n=1$$, tad  ${3}^{1}\equiv i\phantom{\rule{0.147em}{0ex}}\left(\mathit{mod}\phantom{\rule{0.147em}{0ex}}7\right)$
Ja $$n=2$$, tad  ${3}^{2}\equiv 9\equiv i\phantom{\rule{0.147em}{0ex}}\left(\mathit{mod}\phantom{\rule{0.147em}{0ex}}7\right)$
Ja $$n=3$$, tad  ${3}^{3}\equiv {3}^{2}\cdot 3\equiv 2\cdot 3\equiv i\phantom{\rule{0.147em}{0ex}}\left(\mathit{mod}\phantom{\rule{0.147em}{0ex}}7\right)$
Ja $$n=4$$, tad  ${3}^{4}\equiv {3}^{3}\cdot 3\equiv 6\cdot 3\equiv i\phantom{\rule{0.147em}{0ex}}\left(\mathit{mod}\phantom{\rule{0.147em}{0ex}}7\right)$
Ja $$n=5$$, tad  ${3}^{5}\equiv {3}^{4}\cdot 3\equiv i\phantom{\rule{0.147em}{0ex}}\left(\mathit{mod}\phantom{\rule{0.147em}{0ex}}7\right)$
Ja $$n=6,$$ tad  ${3}^{6}\equiv i\phantom{\rule{0.147em}{0ex}}\left(\mathit{mod}\phantom{\rule{0.147em}{0ex}}7\right)$

Virkne ${3}^{n} \left(\mathit{mod}\phantom{\rule{0.147em}{0ex}}7\right)$ ir periodiska ar perioda garumu .

2) Nosaki atlikumu!
${3}^{61}\equiv i\phantom{\rule{0.147em}{0ex}}\left(\mathit{mod}\phantom{\rule{0.147em}{0ex}}7\right)$, tātad skaitlis ${3}^{61}$, dalot ar $$7$$, dod atlikumu  .

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