It all depends on the age, condition and the collect ability of the rifle.
The made them from 48 - 62, they are date code stamped on the barrel, the location and chart is on the Remington Society of America.
Made from 1909 - 1936, with a serial no I can tell you a year, from 1921 on there is a barrel code with month/year. Check Remington Society of America...
Serials are no unique, check the Remington Society site for the barrel code info: http://remingtonsociety.com/rsa/questions/barrelcodes
They were only made from 1932-1935, they are date code stamped on the barrel details on the Remington Society of America, Manufactured date link.