You need more info than the serial number. Date and month codes changed in about 1968.
Your serial number most likely was made in about 1958 to 1960 (Guess). My 760 is serial 337XXX and was mfg in 1960. You can go to the Remington Society web pages and identification process with charts are there.
The thing that changed in 1968 was firearms manufactures were required to put serials on even 22 lr many 22 rifles did not have them on it before then. The page is here:
http://remingtonsociety.com/rsa/questions/barrelcodes
(I put it there ;)
Remington rifles can be dated not from a serial number, but by a 2 or 3 LETTER code, stamped on the left side of the barrel about a quarter inch from the frame. Use a magnifier and a good light. Here is a link to the codes which can tell you month and year of production: http://www.remingtonsociety.org/manufacture-dates/
1920
I checked www.oldguns.net and they show the manufacture date of Remington 1903A3 serial no. 3851292 as 1942.
RW isn't part of the serial no, it stands for Remington Works 1915
That is not a serial no of a Remington 1911A1 they stopped in 1945 with 2465139.
1914
1913
1920
1919
late 1919
1920
1915
1912