Reszta z dzielenia

[angelst]

• Obliczyć reszte z dzielenia

\(\displaystyle{ a= 2^{1997}* 3^{427} przez 25}\)

[czekoladowy]

\(\displaystyle{ 2^{10}\equiv2^{10}(mod25)}\)
\(\displaystyle{ 2^{10}\equiv1024(mod25)}\)
\(\displaystyle{ 2^{10}\equiv-1(mod25)}\)
\(\displaystyle{ (2^{10})^{199}\equiv(-1)^{199}(mod25)}\)
\(\displaystyle{ 2^{1990}\equiv-1(mod25)}\)
\(\displaystyle{ 2^{1990}\cdot2^7\equiv-128(mod25)}\)
\(\displaystyle{ 2^{1997}\equiv-3(mod25)}\)
\(\displaystyle{ 2^{1997}\equiv22(mod25)}\)
\(\displaystyle{ 3^{10}\equiv3^{10}(mod25)}\)
\(\displaystyle{ 3^{10}\equiv59049(mod25)}\)
\(\displaystyle{ 3^{10}\equiv-1(mod25)}\)
\(\displaystyle{ (3^{10})^{42}\equiv(-1)^{42}(mod25)}\)
\(\displaystyle{ 3^{420}\equiv1(mod25)}\)
\(\displaystyle{ 3^{420}\cdot3^7\equiv2184(mod25)}\)
\(\displaystyle{ 3^{427}\equiv9(mod25)}\)
\(\displaystyle{ \begin{cases} 2^{1997}\equiv22(mod25)\\ 3^{427}\equiv9(mod25) \end{cases}}\)
\(\displaystyle{ 2^{1997}\cdot3^{427}\equiv22\cdot9(mod25)}\)
\(\displaystyle{ 2^{1997}\cdot3^{427}\equiv198(mod25)}\)
\(\displaystyle{ 2^{1997}\cdot3^{427}\equiv175+23(mod25)}\)
\(\displaystyle{ 2^{1997}\cdot3^{427}\equiv23(mod25)}\)
Takie cóś.

[xanowron]

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