Show that $N$ is not prime

Question

Show that $N = 10101 \dots 101$ is not prime except when $N=101$.

I have tried to do induction but I am unable to get any equation. I even tried to make factors of different N and take pattern but I was unsuccessful.

Answer 1

Hint: think about polynomial factorizations. This $N$ is equal to $$1+10^2+10^4+\cdots+10^{2k-2}=\frac{10^{2k}-1}{10^2-1}$$ Can we factor that numerator in other ways?