Question

[Lạnh băng kok]

Chứng minh rằng:

\(2005^{2007}+2007^{2005}⋮2006\)

Answer 1

[Shinichi Kudo]

Ta có: 2005²⁰⁰⁷ + 2007²⁰⁰⁵ = (2005²⁰⁰⁷ + 1²⁰⁰⁷ ) + (2007²⁰⁰⁵ - 1²⁰⁰⁵ )

Vì \(2005^{2007}+1^{2007}\)luôn chia hết cho \(2005+1=2006\left(1\right)\)

    \(2007^{2005}-1^{2005}\)luôn chia hết cho \(2007-1=2006\left(2\right)\)

\(\left(1\right)\left(2\right)\Rightarrow\left(2005^{2007}+1^{2007}\right)+\left(2007^{2005}-1^{2005}\right)⋮2006\)

                 \(\Rightarrow2005^{2007}+2007^{2005}⋮2006\)

Vậy \(2005^{2007}+2007^{2005}⋮2006\)