Finding the remainder when $5^{55}+3^{55}$ is divided by $16$ 
Find the remainder when $5^{55}+3^{55}$ is divided by $16$.

What I try
$a^{n}+b^{n}$ is divided by $a+b$ when $n\in $ set of odd natural number.
So $5^{55}+3^{55}$ is divided by $5+3=8$
But did not know how to solve original problem
Help me please

Continuing your method:
$$5^{55}+3^{55}=(5+3)(5^{54}-5^{53}\cdot 3+5^{52}\cdot 3^2-\cdots +5^2\cdot 3^{52}-5\cdot 3^{53}+3^{54}) \Rightarrow \\
\small 5^{54}-5^{53}\cdot 3+5^{52}\cdot 3^2-\cdots +5^2\cdot 3^{52}-5\cdot 3^{53}+3^{54}\equiv 1-1+1-\cdots+1-1+1\equiv 1\pmod{2}$$
Hence:
$$5^{55}+3^{55}\equiv 8 \pmod{16}$$
