Question

[Minh Hoang Hai]

CMR : 222³³³ + 333²²² chia hết cho 13

Answer 1

[Hoang Hung Quan]

Ta có:

\(222^{333}+333^{222}=111^{333}.2^{333}+111^{222}.3^{222}\)

\(=111^{222}\left[\left(111.2^3\right)^{111}+\left(3^2\right)^{111}\right]\)

\(=111^{222}\left(888^{111}+9^{111}\right)\)

\(\Rightarrow888^{111}+9^{111}\)

\(=\left(888+9\right)\left(888^{110}-888^{109}.9+...-888.9^{109}+9^{110}\right)\)

\(=13.69.\left(888^{110}-888^{109}.9+...-9^{109}+9^{110}\right)\)

\(=13.69.Q\)

\(\Rightarrow222^{333}+333^{222}⋮13\) (Đpcm)