[tex]31_4+13_4[/tex]
Adding the digits in the "ones" place gives 4, which in base-4 amounts to setting equal to 0 and carrying the "one" to the "fours" place:
[tex]1_4+3_4=4_4=10_4[/tex]
The same thing happens in the "fours" place, so you end up carrying the "one" to the "sixteens" place.
[tex]30_4+10_4=40_4=100_4[/tex]
So putting this together, you should get
[tex]31_4+13_4=110_4[/tex]
