The old company ledger available on this web site starts at September 30, 1919, at serial number 172939.
http://www.remingtonsociety.com/rsa/questions/FR04
The company name changed from Remington Arms Co. to Remington Arms-Union Metallic Cartridge Co. in 1911. So, I'd guess your gun at not too long after 1911. Your gun looks like a No. 10S "Trap Special" Grade as pictured here from the Rem-UMC 1913-1914 catalogue --
Looks like a real nice example you have, congrats!