TITLE: To show that Fermat number $F_{5}$ is divisible by $641$.
QUESTION [8 upvotes]: How can I show that Fermat number $F_{5}=2^{2^5}+1$ is divisible by $641$.

REPLY [9 votes]: $\!\!\bmod\, 641\!:\,\  {-}1\, \equiv\:\, 5\,\cdot\, 2^{\large 7},\,$  so by Power & Product Rules
$\rm above^{\large 4}\,  \Longrightarrow\ 1^{\phantom{|^{|^|}}}\!\! \equiv \underbrace{\color{#c00}{5^{\large 4}}\!}_{\Large\!\!\!\!\! \color{#c00}{\equiv\,-2^{\Large 4}}}\:\! 2^{\large 28}\equiv\, -2^{\large 32}\ \ \ \ {\small\bf QED}$