TITLE: Simple number theory problem
QUESTION [6 upvotes]: I found this question in a textbook on number theory:

For which integer c will $\;\displaystyle{\frac{c^6 - 3}{c^2 + 2}}\;$ also be an integer?

I wonder if there is a solution which is not based on trial and error.

REPLY [7 votes]: $\,c^2\!+2\mid \color{#0a0}{c^6\!-3}\,\Rightarrow\,$  mod $\,c^2\!+2\!:\,\ c^2\equiv \color{#c00}{ -2},\ \ \color{}0\equiv \color{#0a0}{(c^2)^3\!-3}\equiv (\color{#c00}{-2})^3\!-\!3 \equiv -11,\ $ so $\ c^2\!+2\mid 11.$