Official solution
1 + 49 + 492
+ ..... + 49125
sum of G.P. = \[{{1.\left( {{{49}^{126}} - 1} \right)} \over {49 - 1}}\]
= \[{{\left( {{{49}^{63}} + 1} \right)\left( {{{49}^{63}} - 1} \right)} \over {48}}\]
Also 4963 - 1
= (1 + 48)63 - 1
= [63C0\[ \times \]1 + 63C1\[ \times \] 48 + 63C2\[ \times \] (48)2 + .... ] - 1
= [1 + 48\[\lambda \]] - 1 = 48\[\lambda \]
So \[{{\left( {{{49}^{63}} - 1} \right)} \over {48}}\] = integer
\[ \therefore \] 4963 + 1 is a factor.
So k = 63.