The greatest positive integer k, for which  ${49}^k+1$ is a factor of the sum ${49}^{125}+{49}^{124}+\mathrm{......}+{49}^2+49+1,$is

$$\begin{gathered} {\left(49\right)}^{125}+{\left(49\right)}^{124}+....+49+1=\frac{{\left(49\right)}^{126}-1}{48} \\ =\frac{\left({\left(49\right)}^{63}+1\right)\left({\left(49\right)}^{63}-1\right)}{48} \\ \Rightarrow {\left(49\right)}^{63}+1 is a factor of the the sum. \\ \Rightarrow k=63 \end{gathered}$$