How would you show that $37 \times 2^{220} - 14 \times 5^{87} \times 7^{87}$ is divisible by $17$?
I tried finding the actual answer of the equation to show that it is divisible by $17$, but I think I need a more efficient and stronger proof. What could this proof be?
Hint:
$$2^{4} \equiv -1 \mod 17$$
$$35 \equiv 1 \mod 17$$
Edit:
\begin{align} 37 \times 2^{220} - 14 \times 5^{87} \times 7 ^{87} &\equiv (2 \times 17+3) \times(2^4)^{55}-(17-3) \times 35^{87}\mod 17 \\ &\equiv3 \times (-1)^{55} - (-3)\times 1^{87}\mod17 \\ &\equiv (-3)-(-3) \mod 17 \\ &\equiv 0 \mod 17 \end{align}
