The question given to me is:

> 
> Show that $\large\frac{(3^{77}-1)}{2}$ is odd and composite.
> 
> 
> 

We can show that $\forall n\in\mathbb{N}$:

$$3^{n}\equiv\left\{ 
 \begin{array}{l l}
 1 & \quad \text{if $n\equiv0\pmod{2}$ }\\
 3 & \quad \text{if $n\equiv1\pmod{2}$}\\
 \end{array} \right\} \pmod{4}$$

Therefore, we can show that $3^{77}\equiv3\pmod{4}$. Thus, we can determine that $(3^{77}-1)\equiv2\pmod{4}$. Thus, we can show that $\frac{(3^{77}-1)}{2}$ is odd as:

$$\frac{(3^{77}-1)}{2}\equiv\pm1\pmod{4}$$

However, I am unsure how to show that this number is composite. The book I am reading simply states two of the factors, $\frac{(3^{11}-1)}{2}$ and $\frac{(3^{7}-1)}{2}$, but I do not know how the authors discovered these factors. 

I'd appreciate any help pointing me in the right direction, thanks.