# Find an integer in the range $100\leq n \leq 1997$ such that $\frac{2^n+2}{n}$ is an integer.

I am having trouble with this question. It comes from the 1997 APMO:

Find an integer in the range $$100\leq n \leq1997$$ such that $$\frac{2^n+2}{n}$$ is an integer.

When I first attempted this problem, I thought, maybe FLT (Fermat's Little Theorem) might apply to this. However, I soon found out that $$n$$ must be even, and the only even prime is 2, so FLT is useless. I really don't know what to do. Can someone please help?

• Why do you think that $n$ must be even? – Misha Lavrov Mar 10 at 20:50
• Actually, I just found out that $n$ can be odd, since odd times even = even. Silly me! However, FLT still does not work. I tried to factor out 2 from the numerator and use FLT on the term that is inside, but I still can't seem to make it work. – A R Mar 10 at 20:56
• SPOILER ALERT: mks.mff.cuni.cz/kalva/apmo/asoln/asol972.html – Dr. Mathva Mar 10 at 21:11
• @Dr.Mathva So, then, a question whose answer is readily found using the brute force of mathematical software, but which nobody could reasonably be expected to find under exam conditions. – Rosie F Mar 11 at 15:22