The value of the greatest positive integer $k$,such that $49^k + 1$ is a factor of $48(49^{125} + 49^{124} + \ldots + 49^2 + 49 + 1)$ is

The remainder when $(11)^{1011} + (1011)^{11}$ is divided by $9$ is

The remainder,when $(15^{23} + 23^{23})$ is divided by $19$,is

Given below are two statements:
Statement $I$: $25^{13} + 20^{13} + 8^{13} + 3^{13}$ is divisible by $7$.
Statement $II$: The integral part of $(7 + 4\sqrt{3})^{25}$ is an odd number.
In the light of the above statements, choose the correct answer from the options given below:

Let the number $(22)^{2022} + (2022)^{22}$ leave the remainder $\alpha$ when divided by $3$ and $\beta$ when divided by $7$. Then $(\alpha^2 + \beta^2)$ is equal to

The $50^{\text{th}}$ root of a number $x$ is $12$ and the $50^{\text{th}}$ root of another number $y$ is $18$. Then the remainder obtained on dividing $(x + y)$ by $25$ is $........$.