Numbers in base 4 add up to 4 instead of 10, then first adding the unit we have
[tex]3+3=6=4+2\Rightarrow12_{four}[/tex]Now, adding the next place to the left(considering the unit from the previous addition)
[tex]1+1+2=4\Rightarrow10_{four}[/tex]Then
[tex]13_{\text{four}}+23_{\text{four}}=12_{\text{four}}+30_{\text{four}}=102_{\text{four}}[/tex]