How does the router synchronize time?

The router pulls it's time like any computer. It uses the Network Time Protocol or (NTP). In order to utilize the NTP a valid Internet connection must be achieved.

Network Time Protocol (NTP) is a protocol that is used to synchronize computer clock times in a network of computers. NTP is now an Internet standard. In common with similar protocols, NTP uses Coordinated Universal Time (UTC) to synchronize computer clock times to a millisecond, and sometimes to a fraction of a millisecond.

UTC time is obtained using several different methods, including radio and satellite systems. Specialized receivers are available for high-level services such as the Global Positioning System (GPS) and the governments of some nations. However, it is not practical or cost-effective to equip every computer with one of these receivers. Instead, computers designated as primary time servers are outfitted with the receivers and they use protocols such as NTP to synchronize the clock times of networked computers.

Degrees of separation from the UTC source are defined as strata. A radio clock (which receives true time from a dedicated transmitter or satellite navigation system) is stratum-0; a computer that is directly linked to the radio clock is stratum-1; a computer that receives its time from a stratum-1 computer is stratum-2, and so on. The term NTP applies to both the protocol and the client/server programs that run on computers.

The programs are compiled by the user as an NTP client, NTP server, or both. In basic terms, the NTP client initiates a time request exchange with the time server. As a result of this exchange, the client is able to calculate the link delay, its local offset, and adjust its local clock to match the clock at the server's computer. As a rule, six exchanges over a period of about five to 10 minutes are required to initially set the clock. Once synchronized, the client updates the clock about once every 10 minutes, usually requiring only a single message exchange. redundant servers and varied network paths are used to ensure reliability and accuracy. In addition to client/server synchronization, NTP also supports broadcast synchronization of peer computer clocks. NTP is designed to be highly fault-tolerant and scalable.