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Suppose that

$ABC_4+200_{10}=ABC_9$
ABC_4+200_{10}=ABC_9,
where A, B, and C are valid digits in base 4 and 9.
What is the sum when you add all possible values of A, all possible values of B, and all possible values of C?

1.

A, B, and C are valid digits in base 4 and 9

$A = \{ 0,1,2,3 \} \\ B = \{ 0,1,2,3 \} \\ C = \{ 0,1,2,3 \}$

2.

$\begin{array}{|rcll|} \hline ABC_4+200_{10} &=& ABC_9 \\ \overbrace{A\cdot 4^2 + B\cdot 4 + C}^{ABC_4=} + 200 &=& \overbrace{ A\cdot 9^2 + B\cdot 9 + C}^{ABC_9=} \\ 16A+4B+200 &=& 81A+9B \\ 65A+5B &=& 200 \quad & | \quad :5\\ \mathbf{13A+B} &\mathbf{=}&\mathbf{40} \\ \hline \end{array}$

3.

Possible values of A and B

$\begin{array}{|c|c|r|r|} \hline A & B & 13A+B & =40\ ? \\ \hline 0 & 0 & 0 \\ & 1 & 1 \\ & 2 & 2 \\ & 3 & 3 \\ \hline 1 & 0 & 13 \\ & 1 & 14 \\ & 2 & 15 \\ & 3 & 16 \\ \hline 2 & 0 & 26 \\ & 1 & 27 \\ & 2 & 28 \\ & 3 & 29 \\ \hline 3 & 0 & 39 \\ & 1 & 40 & \checkmark \\ & 2 & 41 \\ & 3 & 42 \\ \hline \end{array}\\ A=3,\ B=1,\ C=0,1,2,3$

$\text{sum} = 3+1+0+1+2+3 = \mathbf{10}$

The sum when you add all possible values of A, all possible values of B, and all possible values of C is 10

check:

$310_4+200_{10} = 310_9 =252_{10} \\ 311_4+200_{10} = 311_9 =253_{10} \\ 312_4+200_{10} = 312_9 =254_{10} \\ 313_4+200_{10} = 313_9 =255_{10}$