How do you divide number written base 3 by 2 without changing base?

Just like in base 10 (but remember that carries between place value columns are in powers of 3 which look like '10' (but in base 3); '10' in base 3 is '3' in base 10; and each place value can only contain a digit in the range 0 to 2).

The 2 times table in base 3 also helps:

1 x 2 = 2
2 x 2 = 11
10 x 2 = 20
11 x 2 = 22
12 x 2 = 101
20 x 2 = 110
...

example 2120122₃ ÷ 2₃:

.... 1021211
.. -------------
2 | 2120122
.... 2
.... --
..... 012
....... 11
....... ---
......... 10
........... 2
.......... --
........... 11
........... 11
........... ---
............. 02
............... 2
.............. --
............... 02
................. 2
............... ---
................. 0

so 2120122₃ ÷ 2₃ = 1021211₃

Checking in base 10:

2120122₃ = 1880₁₀
2₃ = 2₁₀
1021211₃ = 940₁₀

1880₁₀ ÷ 2₁₀ = 940₁₀ (as required)