Question

Chứng tỏ rằng:

\(^{5^{2017}+5^{2016}+5^{2015}}\) chia hết cho 31

Giúp mk với các bạn

Answer 1

5²⁰¹⁷ + 5²⁰¹⁶ + 5²⁰¹⁵ = 5²⁰¹⁵ x ( 5² + 5 + 1) = 5²⁰¹⁵ x (25 + 6) = 5²⁰¹⁵ x 31

Vậy 5²⁰¹⁷ + 5²⁰¹⁶ + 5²⁰¹⁵ chia hết cho 31.

Answer 2

Ta có:  \(5^3\equiv1\left(mod31\right)\)

=> \(\left(5^3\right)^{671}\equiv1\left(mod31\right)\)

=> \(\begin{cases}\left(5^3\right)^{671}\cdot5^2\equiv25\left(mod31\right)\equiv25\left(mod31\right)\\\left(5^3\right)^{671}\cdot5^3\equiv5^3\left(mod31\right)\equiv1\left(mod31\right)\\\left(5^3\right)^{671}\cdot5^3\cdot5\equiv5^4\left(mod31\right)\equiv5\left(mod31\right)\end{cases}\)

=> \(\begin{cases}5^{2015}\equiv25\left(mod31\right)\\5^{2016}\equiv1\left(mod31\right)\\5^{2017}\equiv5\left(mod31\right)\end{cases}\)

=> \(5^{2015}+5^{2016}+5^{2017}\equiv25+5+1\left(mod31\right)\equiv0\left(mod31\right)\)

Vậy \(5^{2015}+5^{2016}+5^{2017}⋮31\left(đpcm\right)\)

Answer 3

5²⁰¹⁷+5²⁰¹⁶+5²⁰¹⁵

5^2015.(5^2+5+1)

5^2015.31 chia hết cho 31 

=> Tổng trên chia hết cho 31 

 

Answer 4

\(5^{2017}+5^{2016}+5^{2015}=5^2.5^{2015}+5.5^{2015}.1.5^{2015}=5^{2015}.\left(5^2+5+1\right)=5^{2015}.31⋮31\) 

\(\Rightarrow5^{2017}+5^{2016}+5^{2015}⋮31\)