Ancient Rome became a republic in

Century numbering groups years into 100-year blocks. For BC years: 600 BC-501 BC is the 6th century BC, and 300 BC-201 BC is the 3rd century BC. For AD years: 1 AD-100 AD is the 1st century AD, and 101 AD-200 AD is the 2nd century AD.

Rome's monarchy ended and the Roman Republic was established around 509 BC, traditionally dated to the overthrow of the last Roman king, Lucius Tarquinius Superbus, and the creation of an annually elected pair of consuls in place of a king. The year 509 BC falls within the range 600 BC-501 BC, so it is placed in the 6th century BC.

Checking the other offered spans against 509 BC confirms none of them contains it:

• 3rd century BC (300 BC-201 BC) is roughly two centuries after 509 BC.

• 1st century AD (1 AD-100 AD) is roughly five centuries after 509 BC, on the other side of the BC/AD divide.

• 2nd century AD (101 AD-200 AD) is later still, roughly six centuries after 509 BC.

Only the 6th century BC contains 509 BC, the founding year of the Roman Republic.