The sum $1! + 4! + 7! + 10! + 12! + 13! + 15! + 16! + 17!$ is divisible by

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:

For any natural number $n$,$(15 \times 5^{2n}) + (2 \times 2^{3n})$ is divisible by

The least remainder when $17^{30}$ is divided by $5$ is

If $a, b$ and $n$ are natural numbers,then $a^{2n-1} + b^{2n-1}$ is divisible by

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 $........$.