To find the greatest positive integer k for which 49k - 1 is a factor of the sum 49125 + 49124 + ... + 492 + 49 + 1, we can use the concept of geometric series.

Geometric series:
A geometric series is the sum of the terms in a geometric sequence. In a geometric sequence, each term is found by multiplying the previous term by a constant. The sum of a geometric series can be calculated using the formula:

S = a * (r^n - 1) / (r - 1)

where S is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms.

Finding the common ratio:
In this case, the common ratio is 49. We can see that each term in the series is obtained by multiplying the previous term by 49.

Finding the number of terms:
The number of terms in the series can be calculated by finding the exponent, n, in the formula:

S = a * (r^n - 1) / (r - 1)

In this case, the sum of the series is given as 49125 + 49124 + ... + 492 + 49 + 1. We can see that the first term, a, is 49125 and the common ratio, r, is 49. We need to find the value of n.

Using the formula, we can rearrange it to solve for n:

S * (r - 1) = a * (r^n - 1)
(r^n - 1) = (S * (r - 1)) / a
n = log(base r) [(S * (r - 1)) / a] + 1

Calculating the sum of the series:
Using the formula for the sum of a geometric series, we can calculate the sum of the given series:

S = 49125 + 49124 + ... + 492 + 49 + 1
S = 49125 * (49^k - 1) / (49 - 1)

Finding the value of k:
Now we can substitute the values of a, r, and S into the formula for n:

n = log(base 49) [(S * (49 - 1)) / 49125] + 1

To find the greatest positive integer k, we need to find the largest integer value for n. We can start by substituting k = 1 into the formula and see if the result is an integer. If it is, we can try larger values of k until we reach a non-integer result.

After substituting k = 1 into the formula, we find that n = 3. Since this is an integer, we can try larger values of k. Continuing this process, we find that k = 63 gives us an integer value for n. Trying larger values of k results in non-integer values for n.

Therefore, the greatest positive integer k for which 49k - 1 is a factor of the sum 49125 + 49124 + ... + 492 + 49 + 1 is 63. Hence, the correct answer is option D.