If $6^{83}+8^{83}$ is divided by 49, the remainder is 

$6^{83}+8^{83}=(7-1{)}^{83}+(7+1{)}^{83}$

              $\begin{array}{l}=2\left(7^{83}{+}^{83}{C}_2\cdot 7^{81}{+}^{83}{C}_4\cdot 7^{79}+\dots {+}^{83}{C}_{30}{7}^3{+}^{83}{C}_{42}7\right)\\ =2\left\{49m{+}^{83}{C}_{32}\cdot 7\right\}\end{array}$

where, $m$ is an integer

$\begin{array}{l}=98m+2{\cdot }^{.3}{C}_1\cdot 7=98m+2\cdot 83\cdot 7\\ =98m+2(77+6)\cdot 7=49(2m+22)+84\end{array}$

$\begin{array}{l}=49(2m+22)+49+35\\ =49(2m+23)+35=49n+35\end{array}$

where n is an integer.

Hence, the remainder is 35.