If $n \in N$,then ${11^{n + 2}} + {12^{2n + 1}}$ is divisible by

If $2 \cdot 4^{2n+1} + 3^{3n+1}$ is divisible by $k$ for all $n \in N$,then $k=$

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:

The difference between an integer and its cube is divisible by

The remainder when $64^{32^{32}}$ is divided by $9$ is equal to .........................

The numbers $a_n = 6^n - 5n$ for $n = 1, 2, 3, \ldots$ when divided by $25$ leave the remainder: