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number theory intermediate
Problem
Find the product of . Express your answer in base 2.
Solution
We carry out the multiplication as we carry out multiplication in base . Fortunately, we do not have to worry about carrying over, since we are only multiplying by digits of or . Thus: \begin{array}{@{}c@{\;}c@{}c@{}c@{}c@{}c@{}c}
& & & 1 & 0 & 1 & 1_2 \\
& & & \times & 1 & 0 & 1_2 \\
\cline{4-7} & & & 1 & 0 & 1 & 1_2 \\
& & 0 & 0 & 0 & 0 & 0_2 \\
+ & 1 & 0 & 1 & 1 & 0 & 0_2 \\ \cline{1-7}
& 1 & 1 & 0 & 1 & 1 & 1_2 \\
\end{array}When summing, we need to carry-over for the second digit from the left. Thus, the sum is equal to .
Final answer
110111