${{3}^{2}}\overset{17}{\mathop{\equiv }}\,9\overset{17}{\mathop{\equiv }}\,-8\xrightarrow{2}{{3}^{4}}\overset{17}{\mathop{\equiv }}\,64-4\times 17\overset{17}{\mathop{\equiv }}\,-4\xrightarrow{2}{{3}^{8}}\overset{17}{\mathop{\equiv }}\,16\overset{17}{\mathop{\equiv }}\,-1$ $\xrightarrow{2}{{3}^{16}}\overset{17}{\mathop{\equiv }}\,1\overset{17}{\mathop{\equiv }}\,1+17\overset{17}{\mathop{\equiv }}\,18\xrightarrow[(3,17)=1]{\div 3}{{3}^{15}}\overset{17}{\mathop{\equiv }}\,6$ ${{3}^{15}}+a\overset{17}{\mathop{\equiv }}\,6+a\overset{17}{\mathop{\equiv }}\,0\Rightarrow {{a}_{\min }}=11$