Prime factors of a number in the form $2^n+1$

Question

What is the gcd of $2^{55} +1$ and $165$?

This question was asked in KVPY 2019 SA.

Answer 1

Hint:

$165=5\times3\times11$

$2^{55}\equiv2^3\equiv3\pmod5$

$2^{55}\equiv2\equiv-1\pmod3$

$2^{55}\equiv2^5\equiv-1\pmod{11}$