Question

chứng minh rằng\(^{3^{15}-^{ }9^6}\)chia hết cho 13

Answer 1

\(3^{15}-9^6=3^{15}-3^{12}=3^{12}\left(3^3-1\right)=3^{12}\cdot26\)

Vì: \(26⋮13\Rightarrow3^{12}\cdot26⋮13\Rightarrow3^{15}-9^6⋮13\)

--> Đpcm

Answer 2

\(3^{15}-9^6\)

\(=3^{15}-\left(3^2\right)^6\)

\(=3^{15}-3^{12}\)

\(=3^{12}.3^3-3^{12}.1\)

\(=3^{12}.27-3^{12}.1\)

\(=3^{12}\left(27-1\right)\)

\(=3^{12}.26\)

\(=3^{12}.2.13⋮13\Rightarrowđpcm\)

Answer 3

Ta có: \(3^3\equiv1\left(mod13\right)\)

\(\Rightarrow3^{15}\equiv1\left(mod13\right)\)

\(9^3\equiv1\left(mod13\right)\)

\(\Rightarrow9^6\equiv1\left(mod13\right)\)

\(\Rightarrow3^{15}-9^6\equiv0\left(mod13\right)\)

\(\Rightarrow3^{15}-9^6⋮13\)

\(\Rightarrowđpcm\)