\(\left(222^{333}+333^{222}\right)⋮13\)
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a) \(222^{333}+333^{222}\)
\(=\left(111.2\right)^{333}+\left(111.3\right)^{222}\)
\(=111^{333}.2^{333}+111^{222}.3^{222}\)
\(=111^{222}.\left(111^{111}.2^{333}+3^{222}\right)\)
\(=111^{222}.\left(111^{111}.2^{3.111}+3^{2.111}\right)\)
\(=111^{222}.\left[111^{111}.\left(2^3\right)^{111}+\left(3^2\right)^{111}\right]\)
\(=111^{222}.\left(111^{111}.8^{111}+9^{111}\right)\)
\(=111^{222}.\left[\left(111.8\right)^{111}+9^{111}\right]\)
\(=111^{222}.\left(888^{111}+9^{111}\right)\)
\(=111^{222}.\left(888+9\right)\left[888^{110}-888^{109}.9+.....-888.9^{109}+9^{110}\right]\)
\(=111^{222}.7992\left[888^{110}-888^{109}.9+.....-888.9^{109}+9^{110}\right]\)
\(=111^{222}.897\left[888^{110}-888^{109}.9+.....-888.9^{109}+9^{110}\right]\)
\(=111^{222}.13.69\left[888^{110}-888^{109}.9+.....-888.9^{109}+9^{110}\right]⋮13\)
Vậy \(222^{333}+333^{222}⋮13\left(dpcm\right)\)
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