Question

Chứng minh rằng:

2015²⁰¹⁵-1 chia hết cho 2014

Answer 1

Ta có :\(2015\equiv1\left(mod2014\right)\)

\(\Rightarrow2015^{2015}\equiv1\left(mod2014\right)\)

\(\Rightarrow2015^{2015}-1\equiv0\left(mod2014\right)\)

hay : \(2015^{2015}-1⋮2014\) (đpcm)

Answer 2

\(2015^{2015}-1=2015^{2015}-2015^{2014}+2015^{2014}-2015^{2013}+.....+2015-1\)

\(=\left(2015^{2015}-2015^{2014}\right)+\left(2015^{2014}-2015^{2013}\right)+....+\left(2015-1\right)\)

\(=2015^{2014}.\left(2015-1\right)+2015^{2013}.\left(2015-1\right)+....+\left(2015-1\right)\)

\(=2014.\left(2015^{2014}+2015^{2013}+...+1\right)⋮2014\)