Đặt [tex]4^{101}=68m+n(m,n\in \mathbb{N};0\leq n\leq 68)[/tex]
Dễ thấy: [tex]4^{101}\vdots 4\Rightarrow 68m+n\vdots 4\Rightarrow n\vdots 4\Rightarrow n\in \left \{ 0;4;8;...;64 \right \}[/tex]
Lại có: [tex]4^{101}=(4^{4})^{25}.4=256^{25}.4=(17.15+1)^{25}.4[/tex] chia 17 dư 4. [tex]\Rightarrow 68m+n-4\vdots 17\Rightarrow n-4\vdots 17[/tex]
[tex]\Rightarrow \left\{\begin{matrix} n\vdots 4\\ n-4\vdots 17 \end{matrix}\right.\Rightarrow \left\{\begin{matrix} n-4\vdots 4\\ n-4\vdots 17 \end{matrix}\right.\Rightarrow n-4\vdots 68\Rightarrow n-4=0\Rightarrow n=4\Rightarrow 4^{101}=68m+4\Rightarrow 4^{101}[/tex] chia 68 dư 4.
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