The 583.880 is not the serial #, that is the Sears part number. The serial number will be on the other side of the frame.
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Is it a revolver or semi-auto? You can go to Ruger's website to look up the serial number.
Modle 800 what - rifle, pistol, shotgun?
Serial numbers are not normally found on the barrel, since by law the receiver (frame) is the gun- and a barrel is just "parts". By any chance is your 30-06 marked "J C Higgins" or "Ted Williams"? If so, that number is the Sears Roebuck model number. Many of the Sears branded rifles and shotguns do not HAVE a serial number- they were not required by US law until 1968. And yes, it is perfectly legal to own a gun that never had a serial number. REMOVING a serial number is not legal.
Depends on condition & whether all major parts have the same serial number stamped on it & when in 1943 it was produced. A poor example with non matching serial numbers can go for 300.00 or so. A pristine example with 98-100 % finish, matching serial numbers etc can go for 800.00 or more. A WW2 P38 holster can add 100-200 to the price.
The X signifies that it is a duplicate serial numberr