Model 581.
First year 1909
1919 depends entirely on condition $100-1,200
Your Remington model 1903-A3 was made in the year 1942.It was chambered in the caliber 30-06.
You will have to contact Remington.
1 in 10
List the caliber and then someone will try to help you.
Impossible to tell unless you have the factory records.
$500 up
1920
Are you sure of the model number?
SN's were not required before 1968