They made over 50,000 of them from 1890-1933 in 22, 25, and 32. The octagon barrels are most common with the rounded being later production. Take-down models were available after the turn of the century in fewer numbers and can bring a small premium. The only real collector interest and value is in excellent to mint condition and they bring $400-500.-
You will have to contact Remington.
Your Remington model 1903-A3 was made in the year 1942.It was chambered in the caliber 30-06.
That's a very late gun probably 1933, see related links for date code stamp.
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The Remington model 4 rolling block rifle was made between 1890-1933.If you have a model 4S military rifle,then these were made by Remington from 1913-1923.your rifle has either a 22.5in barrel,or a 24in barrel.These were chambered in .22,.25 Stevens,or 32 caliber.
List the caliber and then someone will try to help you.
Impossible to tell unless you have the factory records.
Model 581.
$500 up
1920
Are you sure of the model number?
SN's were not required before 1968