a) \(21^{10}-1=\left(21^5\right)^2-1^2=\left(21^5+1\right).\left(21^5-1\right)\)
\(21^5+1=\overline{...1}=2k+1+1=2n\)
\(21^5-1=\overline{...01}-1=\overline{...00}\)
\(\Rightarrow21^{10}-1=2n.\overline{...00}⋮200\left(đpcm\right).\)
b) \(39\equiv-1\left(mod40\right)\)
\(\Rightarrow39^{20}\equiv1\left(mod40\right)\)
\(\Rightarrow39^{19}\equiv-1\left(mod40\right)\)
\(\Rightarrow39^{20}+39^{19}\equiv1+\left(-1\right)\left(mod40\right)\)
\(\Leftrightarrow39^{20}+39^{19}\equiv0\left(mod40\right)\)
\(\Rightarrow39^{20}+39^{19}⋮40\left(đpcm\right).\)
d) \(2005\equiv-1\left(mod2006\right)\)
\(\Rightarrow2005^{2007}\equiv\left(-1\right)^{2007}=-1\left(mod2006\right)\)
\(2007\equiv1\left(mod2006\right)\)
\(\Rightarrow2007^{2005}\equiv1\left(mod2006\right)\)
\(\Rightarrow2005^{2007}+2007^{2005}\equiv-1+1=0\left(mod2006\right)\)
\(\Leftrightarrow2005^{2007}+2007^{2005}⋮2006\left(đpcm\right).\)
