Question

a) cho A = 5 + 5² + 5^{3 +}5⁴+...+5¹⁹+5²⁰

     chứng tỏ rằng a :hết cho 6 : a chia hết cho 13

 

Answer 1

ghép mỗi cặp là 2 thì chia hết đấy là phần a

ghép mỗi cặp là 4 thì chia hết đấy là phần b

Answer 2

\(A=5+5^2+5^3+...+5^{19}+5^{20}\)

\(A=\left(5+5^2\right)+\left(5^3+5^4\right)+...+\left(5^{19}+5^{20}\right)\)

\(A=1\cdot\left(5+5^2\right)+5^2\cdot\left(5+5^2\right)+...+5^{18}\cdot\left(5+5^2\right)\)

\(A=1\cdot30+5^2\cdot30+...+5^{18.}\cdot30\)

\(A=30\cdot\left(1+5^2+...+5^{18}\right)\)

\(A=5\cdot6\cdot\left(1+5^2+...+5^{18}\right)⋮6\)

\(A⋮6\)

\(A=5+5^2+5^3+...+5^{19}+5^{20}\)

\(A=\left(5+5^2+5^3+5^4\right)+\left(5^5+5^6+5^7+5^8\right)+...+\left(5^{17}+5^{18}+5^{19}+5^{20}\right)\)

\(A=5\cdot\left(1+5+5^2+5^3\right)+5^5\cdot\left(1+5+5^2+5^3\right)+...+5^{17}\cdot\left(1+5+5^2+5^3\right)\)

\(A=5\cdot156+5^5\cdot156+...+5^{17}\cdot156\)

\(A=156\cdot\left(5+5^2+...+5^{17}\right)\)

\(A=12\cdot13\cdot\left(5+5^2+...+5^{17}\right)⋮13\)

\(A⋮13\)