\( {7}^{11}\)除以6的余数是___．

∵\({7^{11}} = {\left( {1 + 6} \right)^{11}}\)

根据二项展开式不妨设：\({\left( {1 + 6} \right)^{11}} = {a_0} + {a_1} \times 6 + {a_2} \times {6^2} + ... + {a_{11}} \times {6^{11}}\)

显然 \({a_1} \times 6 + {a_2} \times {6^2} + ... + {a_{11}} \times {6^{11}}\) 可被6整除且 \({a_0} = {\rm{C}}_{11}^0 \times {1^{11}} \times {6^0} = 1\)

711除以6的余数是 \({a_0} = 1\)

故答案为：1．

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