<s>[INST] 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
 [/INST] We can simplify the expression as follows: $5^{55} + 3^{55} \equiv (5^4)^{13}\cdot 5^3 + (3^4)^{13}\cdot 3^3 \pmod {16}\\
\equiv (9^2)^{13}\cdot 5^2\cdot 5 + 1^{13}\cdot 11\pmod {16}\\
\equiv 1^{13}\cdot9\cdot 5 + 11\pmod {16}\\
\equiv 8\pmod {16}$. 
So the remainder is $8$.
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