Suppose that \(ABC_4+200_{10}=ABC_9\), 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?
+26376 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:
\(310_4+200_{10} = 310_9 =252_{10} \\ 311_4+200_{10} = 311_9 =253_{10} \\ 312_4+200_{10} = 312_9 =254_{10} \\ 313_4+200_{10} = 313_9 =255_{10} \)
