Question

Chứng tỏ rằng 1+ 5 + 5^2 +.........+ 5^2021 chia hết cho 31

Answer 1

\(1+5+5^2+...+5^{2021}\)

\(=\left(1+5+5^2\right)+\left(5^3+5^4+5^5\right)+...+\left(5^{2019}+5^{2020}+5^{2021}\right)\)

\(=1\left(1+5+5^2\right)+5^3\left(1+5+5^2\right)+...+5^{2019}\left(1+5+5^2\right)\)

\(=1.31+5^3.31+...+5^{2019}.31\)

\(=31.\left(1+5^3+...+5^{2019}\right)⋮31\)

Vậy \(1+5+5^2+...+5^{2019}⋮31\)

Answer 2

(1+5+5²+...+5²⁰²¹)⋮31

<=>[(1+5+5²)+(5³+5⁴+5⁶)+....+(5²⁰¹⁹+5²⁰²⁰+5²⁰²¹)]⋮31

<=>[1.31+5³.(1+5+5²)+...+5²⁰¹⁹.(1+5+5²)]⋮31

<=>(1.31+5³.31+...+5²⁰¹⁹.31)⋮31

<=>[31.(1+5³+....+5²⁰¹⁹)]⋮31

Vì 31⋮31 nên 1+5+5²+...+5²⁰²¹⋮31