division_algorithm_quiz.tex

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\activitytitle{Quiz on things related to the Division Algorithm}{10 points}

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\item Let $n > 0$ and $k > 0$ be integers.
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 $a$ and $b$ for which $n = ak + b$ and $0 \leq b < k$.
Show that $q = a$ and $r = b$.
This shows that there is at most one way to write $n = qk + r$ with $0 \leq r < k$.

\item Let $k$ be an integer.
Show that exactly one of the integers $k, k+1, k+2, k+3$ is a multiple of 4.
You can use the back of this sheet of paper.
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