Evaluate \((111_{two})^2\) and leave your answer in base 2

Mathematics

WAEC 2007

A. \(111001_{two}\)
B. \(110001_{two}\)
C. \(101001_{two}\)
D. \(10010_{two}\)

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Correct Answer: Option B

Explanation

\((111_{two})^{2}=(111_{two})(111_{two})\\
=1 \times 2^{2}+1\times2^{1}+1\times 2^{0}=4+2+1=7_{_{ten}}\\
(111_{two})^{2}=7\times 7=49\\
\begin{matrix}
2 & 49\\
2 & 24\hspace{1mm}R\hspace{1mm}1\\
2 & 12\hspace{1mm}R\hspace{1mm}0\\
2 & 6\hspace{1mm}R\hspace{1mm}0\\
2 & 3\hspace{1mm}R\hspace{1mm}0\\
2 & 1\hspace{1mm}R\hspace{1mm}1\\
& 0\hspace{1mm}R\hspace{1mm}1\hspace{1mm}\uparrow
\end{matrix} \\
=110001_{2}\)

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