Question

 

Chứng tỏ: C = 5 +5²+5³+...+5¹⁹+5²⁰ chia hết cho 13

Answer 1

\(C=\left(5+5^2+5^3+5^4\right)+\left(5^5+5^6+5^7+5^8\right)...+\left(5^{17}+5^{18}+5^{19}+5^{20}\right)\\ C=5\left(1+5+5^2+5^3\right)+5^5\left(1+5+5^2+5^3\right)...+5^{17}\left(1+5+5^2+5^3\right)\\ C=5\cdot156+5^5\cdot156+...+5^{17}\cdot156\\ C=156\left(5+5^5+...+5^{17}\right)\\ C=12\cdot13\left(5+5^5+...+5^{17}\right)⋮17\)

Answer 2

(5 +5³)+(5²+5⁴)...+(5¹⁸+5²⁰)

5(1+5²)+5²(1+5²)+...+5¹⁸(1+5²)

(1+5²)(5+5²+...+5¹⁸)

26(5+5²+...+5¹⁸)⋮13

vậy (5 +5³)+(5²+5⁴)...+(5¹⁸+5²⁰)⋮13