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

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

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Solution:

$1 + 49 + 492 +\ldots+ 49^{12}$
$= \left(49\right)^{126} - 1 = \left(49^{63} + 1\right) \frac{\left(49^{63}-1\right)}{\left(48\right)}$
So greatest value of $k = 63$