The greatest positive integer k, for which 49^k+1 is a factor of the sum 49^125+49^124++49…

Question

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

Accepted Answer

$$\sum_{i=0}^{125}49^i = \frac{49^{126}-1}{48} = \frac{(49^{63}+1)(49^{63}-1)}{48}.$$

So $49^{63}+1$ is a factor, giving $k=63$.

Solutions · 1