$a^n + b^n$ is divisible by which of the following if $n$ is any odd positive integer?

The expression $(1 + x)^n - nx - 1$ is divisible by (where $n \in N$ and $n > 1$):

For a positive integer $n$, $n^{3} + 2n$ is always divisible by

$(2^{3n} - 1)$ will be divisible by $(\forall n \in N)$

Using the binomial theorem,prove that $6^{n} - 5n$ always leaves a remainder of $1$ when divided by $25$ for all $n \in N$.

If $[x]$ denotes the greatest integer function of $x$,then the number of positive integral divisors of $\left[(2+\sqrt{3})^5\right]$ is