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jmc

number theory intermediate

Problem

Find . Express your answer in base .
Solution
We start subtract the rightmost digits, keeping in mind that we are in base .

Since is less than , we must borrow from the , which then becomes . Since , we have in the rightmost digit. Since the left over is less than , we must borrow from the , which becomes a . , so we have in the second rightmost digit. Since , the third rightmost digit is 0. In the usual notation, this process looks like \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c} & & & \cancelto{2}{3}& \cancelto{1}{2} & 5_8\\ & & - & 2 & 3 & 7_8\\ \cline{2-6} & & & & 6& 6_8\\ \end{array}Therefore, our answer is .
Final answer
66_8