The greatest positive integer \[k\], for which \[{49^k} + 1\] is a factor of the sum \[{49^{125}}\; + {\text{ }}{49^{124}}\; + {\text{ }} \cdots {\text{ }} + {\text{ }}{49^2}\; + {\text{ }}49{\text{ }} + {\text{ }}1\], is
A. 32
B. 60
C. 65
D. 63

Hint: In this question, we need to find the value of greatest positive integer \[k\]. For this, we need to use the concept of geometric progression. We will use the below formula for the sum of GP sequence.

Formula used : The sum of all the terms of a geometric progression is given by
\[S=\dfrac{a\left( {{r}^{n}}-1 \right)}{\left( r-1 \right)};r>1\]
Here, \[a\] is the first term of a sequence.
\[r\] is the common ratio.
\[n\] is the total number of terms in the sequence.

Complete step-by-step solution:
We know that the given sequence is \[{49^{125}}\; + {\text{ }}{49^{124}}\; + {\text{ }} \cdots {\text{ }} + {\text{ }}{49^2}\; + {\text{ }}49{\text{ }} + {\text{ }}1\]
Let us rearrange the terms of the given sequence.
Thus, we get
\[1 + 49 + {49^2} + ..... + {49^{125}}\;\]….. (1)
It is in the form of \[a + ar + a{r^2} + ..... + a{r^n}\;\]…… (2)
So, the given sequence is the geometric progression sequence.
Thus, by comparing the sequence (1) and sequence (2), we get
Here, \[a = 1;r = 49;n = 126\]
Now, we will find the sum of all the terms of GP sequence.
\[S = \dfrac{{a\left( {{r^n} - 1} \right)}}{{\left( {r - 1} \right)}}\]
By putting the values of \[a,r,n\] in the above equation, we get
\[S = \dfrac{{1\left( {{{49}^{126}} - 1} \right)}}{{\left( {49 - 1} \right)}}\]
\[S = \dfrac{{\left( {{{49}^{126}} - 1} \right)}}{{\left( {48} \right)}}\]
\[S = \dfrac{{\left( {{{49}^{63}} - 1} \right)\left( {{{49}^{63}} + 1} \right)}}{{\left( {48} \right)}}\]
But we know that \[{49^k} + 1\] is the factor of the above sum.
By comparing \[\left( {{{49}^{63}} + 1} \right)\] with \[{49^k} + 1\], we get
\[k = 63\]
Hence, the value of greatest positive integer \[k\] is \[63\].

Therefore, the correct option is (D).

Note: Here, students generally make mistakes in finding the sum of all the terms in the GP sequence. While taking the value of \[n\], usually they may get wrong as they consider \[n = 125\] but the total terms in the sequence are \[126\]. Due to this, the required result may get wrong.