The Remington Model 10 A was manufactured between 1908 and 1929, with a total production of about 275,600 units. Many different grades were made, designated by the letter/number as follows:
No. 1 or 10 A - Standard or Field Grade
No. 2 or 10 B - Special Grade
No. 3 or 10 C - Trap Grade
No. 3 Trap Special Grade
No. 4 or 10 D - Tournament Grade
No. 5 or 10 E - Expert Grade
No. 6 or 10 F - Premier Grade
No. 0 Riot Grade
They also made two versions made as well:
Model 10 Trench Shotgun (World War 1)
Model 10 T – Target Grade which included grades D, E, and F as described above
The guns had a simple serial code system, which started with 001 and went to 275,600. The serial number on all guns made was preceded by the letter U.
According to the [Only registered and activated users can see links. ] barrel date codes were used starting in 1921. This means that only later guns would have a date, specifically in the last 8 years of production.
According to the information you provided, the serial number correlates to before September of 1919 (See [Only registered and activated users can see links. ], page 11). Records were recorded for Sept of 1919 showing that the serial numbers for that month were already above 166,000. I also found another source that shows the 1913 serial numbers ranging from 95318 through 115104. It is possible that the gun is from as early as 1914.
It would be a nice collectors gun, but ealier models until around 1912 had issues with the hardness of metals being low causing eventual part failure, which chances are will be hard to find replacement parts for. Also, some guns are chambered for 2 3/4", while some are chambered for 2 9/16" to allow for the shell opening/un-crimping (good old days of paper shells as well). It would be worth having a gunsmith take a very good look at it before you take it out in the field and put some modern shells through it if you were going to shoot it.
If you decide to take it apart do be careful. Quite a complicated design for a pump gun. You can find a [Only registered and activated users can see links. ] here.
CONSIDERABLY more expert information here: [Only registered and activated users can see links. ]