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number theory intermediate

Problem

Calculate the whole number remainder when 987,670 is divided by 128.
Solution
We long divide as follows: \begin{array}{c|c@{\hspace{0pt}}c@{\hspace{0pt}}c@{\hspace{0pt}}c@{\hspace{0pt}}c@{\hspace{0pt}}c@{\hspace{0pt}}c@{\hspace{0pt}}c} \multicolumn{2}{r}{} & & & 7 & 7 & 1 & 6 & \\ \cline{2-8} 128 && 9&8&7&6&7&0 \\ \multicolumn{2}{r}{} & 8 & 9 & 6 &&&& \\ \cline{3-5} \multicolumn{2}{r}{} & & 9 & 1 & 6 &&& \\ \multicolumn{2}{r}{} & & 8 & 9 & 6 &&& \\ \cline{4-6} \multicolumn{2}{r}{} & & & 2 & 0 & 7 && \\ \multicolumn{2}{r}{} & & & 1 & 2 & 8 && \\ \cline{5-7} \multicolumn{2}{r}{} & & & & 7 & 9 & 0 & \\ \multicolumn{2}{r}{} & & & & 7 & 6 & 8 & \\ \cline{6-8} \multicolumn{2}{r}{} & & & & & 2 & 2 & \\ \end{array} So the remainder is .



Assuming use of a calculator, we can divide by 128 to find that the quotient is between 7716 and 7717. Subtracting the product of 7716 and 128 from gives .
Final answer
22