# The greatest positive integer k,

Question:

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

1. (1) 32

2. (2) 63

3. (3) 60

4. (4) 65

Correct Option: , 2

Solution:

$\frac{(49)^{126}-1}{48}=\frac{\left((49)^{63}+1\right)\left(49^{63}-1\right)}{48}\left[\because S_{n}=\frac{a\left(r^{n}-1\right)}{r-1}\right]$

$\therefore \mathrm{K}=63$