Question

Chứng minh 

+)2³¹-1 chia hết cho 7

+)2⁷⁰+3⁷⁰ chia hết cho 13

+)17¹⁹+19¹⁷ chia hết cho18

+)36⁶³-1 chia hết cho 7

+)3⁴ⁿ⁺⁴-4³ⁿ⁺³chia hết cho 17

+)7.5²ⁿ+12.6ⁿ chia hết ch 19

 

Answer 1

phần a sai đề nha bạn 

b,Ta có

      \(2\equiv2\left(mod13\right)\)

\(\Rightarrow2^{12}\equiv1\left(mod13\right)\)

\(\Rightarrow2^{12.5}.2^{10}\equiv1.2^{10}\left(mod13\right)\)

\(\Rightarrow2^{60}.2^{10}\equiv1024\left(mod13\right)\)

\(\Rightarrow2^{70}\equiv10\left(mod13\right)\)\(\left(1\right)\)

Lại có:

\(3\equiv3\left(mod13\right)\)

\(\Rightarrow3^6\equiv1\left(mod13\right)\)

\(\Rightarrow3^{6.11}.3^4\equiv1.3^4\left(mod13\right)\)

\(\Rightarrow3^{66}.3^4\equiv81\left(mod13\right)\)

\(\Rightarrow3^{70}\equiv3\left(mod13\right)\)\(\left(2\right)\)

Từ \(\left(1\right);\left(2\right)\Rightarrow2^{70}+3^{70}\equiv13\equiv0\left(mod13\right)\)

Answer 2

c, Ta có

\(17\equiv-1\left(mod18\right)\)

\(\Rightarrow17^{19}\equiv-1\left(mod18\right)\)\(\left(1\right)\)

Lại có

\(19\equiv1\left(mod18\right)\)

\(\Rightarrow19^{17}\equiv1\left(mod18\right)\)\(\left(2\right)\)

Từ \(\left(1\right);\left(2\right)\Rightarrow17^{19}+19^{17}\equiv0\left(mod18\right)\)

\(\Rightarrow17^{19}+19^{17}⋮18\)

Answer 3

d,Ta có

\(36\equiv1\left(mod7\right)\)

\(\Rightarrow36^{63}\equiv1\left(mod7\right)\)

\(\Rightarrow36^{63}-1\equiv0\left(mod7\right)\)

\(\Rightarrow36^{63}-1⋮7\)