It is a well-known fact that the group homomorphism $\mathrm{SL}(2,\mathbb{Z}) \rightarrow \mathrm{SL}(2,\mathbb{Z}/N\mathbb{Z})$ is surjective.

What I want is a proof by method of algebraic geometry. For example, Let $G= \mathrm{SL}(n,\mathbb{Z})$ be the corresponding group scheme over $\mathbb{Z}$. Then $\mathrm{Spec}(\mathbb{Z}/N) \hookrightarrow \mathrm{Spec}(\mathbb{Z})$ gives $G(\mathbb{Z}) \rightarrow G(\mathbb{Z}/N)$. Then use algebro-geometric method to show that this map is surjective.

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