division_algorithm_quiz_2.tex

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\activitytitle{Quiz on the Division Algorithm}{40 points}

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\item Let $n$ and $k$ be integers and suppose that $k > 0$.
Suppose that there exist integers $q$ and $r$ for which $n = qk + r$ and $0 \leq r < k$, and at the same time that there exist integers $q_2$ and $r_2$ for which $n = q_2 k + r_2$ and $0 \leq r_2 < k$.
Show that $q = q_2$ and $r = r_2$, including deriving any new inequalities that you need.
Bonus points for doing this without dividing by $k$.
This shows that there is at most one way to write $n$ as $qk + r$ with $0 \leq r < k$.

\item Write your solution to this problem on the back of the page.
Let $n$ be even, so that $n = 2k$ for some integer $k$.
Use the Division Algorithm to write $k$ as $2j$ or $2j+1$.
For each case, show that exactly one of the numbers $n$ and $n+2$ is a multiple of 4.
This will also require the use of the Division Algorithm.

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