Go to http://www.remington.com/support/repair_services/ Select Model 10 and the sate you are in. You might have to try adjoining states if no one in your state shows up.
The Remington Model 10 slide action has barrel lengths of 26" 28" 30" and 32"
I just so happen to own a Remington UMC model 10, actually it is pre model 10, they did not name this the model 10 until 1911 I believe. Mine actually has the dates 1903 and 1905 on it. But to get to your answer it stands for UNION METALLIC CARTRIDGE.
According to Remington, it was made in 1913.
1920
1913 (early)
1917
1916
1909
1910
1920
In reference to what application?
1912