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If AAA_4 can be expressed as 33_b, where A is a digit in base 4 and b is a base greater than 5, what is the smallest possible sum A+b?

Guest Aug 14, 2018

#1
+20153
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If AAA_4 can be expressed as 33_b, where A is a digit in base 4 and b is a base greater than 5,

what is the smallest possible sum A+b?

$$\text{AAA_4 = 33_b } \\ \text{Let A = \{0,1,2,3\} }$$

$$\begin{array}{|rcll|} \hline A\cdot 4^2+A\cdot 4^1 + A\cdot 4^0 &=& 3b^1 + 3b^0 \\ A\cdot 16+A\cdot 4 + A &=& 3b + 3 \\ 21A &=& 3\cdot (b + 1) \quad & | \quad : 3 \\ 7A &=& b + 1 \\ \mathbf{b} &\mathbf{=}& \mathbf{7A -1} \quad & | \quad A = \{0,1,2,3\} \\ \hline \end{array}$$

$$\begin{array}{|r|r|c|r|} \hline A & \mathbf{b=7A -1} & b \gt 5 & A+b \\ \hline 0 & -1 & & -1 \\ \hline 1 & 6 & \checkmark & \color{red}7 \\ \hline 2 & 13 & \checkmark & 15 \\ \hline 3 & 20 & \checkmark & 23 \\ \hline \end{array}$$

$$\text{b \gt 5 and the smallest possible sum \mathbf{A+b} is 1+6 \mathbf{=7}  }$$

$$111_4 = 33_6$$

heureka  Aug 14, 2018
#1
+20153
+2

If AAA_4 can be expressed as 33_b, where A is a digit in base 4 and b is a base greater than 5,

what is the smallest possible sum A+b?

$$\text{AAA_4 = 33_b } \\ \text{Let A = \{0,1,2,3\} }$$

$$\begin{array}{|rcll|} \hline A\cdot 4^2+A\cdot 4^1 + A\cdot 4^0 &=& 3b^1 + 3b^0 \\ A\cdot 16+A\cdot 4 + A &=& 3b + 3 \\ 21A &=& 3\cdot (b + 1) \quad & | \quad : 3 \\ 7A &=& b + 1 \\ \mathbf{b} &\mathbf{=}& \mathbf{7A -1} \quad & | \quad A = \{0,1,2,3\} \\ \hline \end{array}$$

$$\begin{array}{|r|r|c|r|} \hline A & \mathbf{b=7A -1} & b \gt 5 & A+b \\ \hline 0 & -1 & & -1 \\ \hline 1 & 6 & \checkmark & \color{red}7 \\ \hline 2 & 13 & \checkmark & 15 \\ \hline 3 & 20 & \checkmark & 23 \\ \hline \end{array}$$

$$\text{b \gt 5 and the smallest possible sum \mathbf{A+b} is 1+6 \mathbf{=7}  }$$

$$111_4 = 33_6$$

heureka  Aug 14, 2018