TITLE: Show that $\frac{(3^{77}-1)}{2}$ is odd and composite
QUESTION [13 upvotes]: 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.

REPLY [26 votes]: Another way to see this is by writing the number in base 3:
$$3^{77}=1\underbrace{00\dots00}_{77}\ _3$$
Here the index $3$ denotes base 3, and $77$ is the number of digits. Subtracting one, we get:
$$3^{77}-1=\underbrace{22\dots22}_{77}\ _3$$
Therefore, dividing this by two,
$$\frac{3^{77}-1}{2}=\underbrace{11\dots11}_{77}\ _3$$
From this we can directly read that the number is odd, since it is the sum of 77 odd numbers, and composite, since $$\underbrace{11\dots11}_{77}\ _3=1111111_3\cdot\underbrace{10000001000000\dots100000010000001}_{71}\ _3$$
(Although, this is basically the same as some of the other answers.)