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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?

 Aug 14, 2018

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 #1
avatar+21848 
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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?

 

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

 

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

 

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

 

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

\(111_4 = 33_6\)

 

laugh

 Aug 14, 2018
 #1
avatar+21848 
+2
Best Answer

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?

 

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

 

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

 

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

 

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

\(111_4 = 33_6\)

 

laugh

heureka Aug 14, 2018

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