Tell whether $\dfrac{10^{91}-1}{9}$ is prime or not? I really have no idea how to start. The only theorem considering prime numbers I know of is Fermat's little theorem and maybe its related with binomial theorem.
Any help will be appreciated.

Just think through the actual number.
$10^{91}$ is a $1$ with $91$ $0$'s after it.
$10^{91}-1$ is therefore $91$ $9$'s in a row.
$\frac{10^{91}-1}{9}$ is therefore $91$ $1$'s in a row.
Due to the form of this number, $x$ $1$'s in a row will divide it, where $x$ is a divisor of $91$.
For example $1111111$ is a divisor, so is $1111111111111$.
Hence the number is not prime.
