when was mexico founded

Mexico as a modern, independent country is usually considered “founded” in 1821 , when it won independence from Spain and emerged as a new sovereign state after three centuries as the Viceroyalty of New Spain.

If you mean the republic specifically (and not the short‑lived Mexican Empire that came right after independence), then a second key “founding” date is October 4, 1824 , when the United Mexican States (the First Mexican Republic) was formally established by constitution.