Show that I is a principal ideal

Question

Let $\mathbb{Z}[\sqrt{-13}]$ be the smallest subring of $\mathbb{C}$ containing $\mathbb{Z}$ and $\sqrt{-13}$ and let the ideal $I = \left<2,\sqrt{-13}\right>$. Show that $I$ is a principal ideal.

I have earnestly tried to prove this problem, but I can't solve it. Is it that there would be someone who could help me at this level?

Answer 1

$x=(2-\sqrt{ -13})(2+\sqrt {-13})=4+13=17\in I$ so $1=17-16=x-8*2\in I$ and $I=\mathbb Z[\sqrt{-13}]$.

Answer 2

Similarly to the other answer: $$1=7*2+\sqrt{-13}\sqrt{-13} \in I$$