The greatest positive integer k, for which 49^k + 1 is a factor of the sum
49¹²⁵ + 49¹²⁴ + ..... + 49² + 49 + 1, is:

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