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Problem 37: (Existence Proof Of The Division Algorithm)

Let $a$ and $b$ be integers with $b>0$. Prove that there exists integers $q$ and $r$ such that $a=qb+r$ and $0\leq r<b$.

Click for a hint.

Consider the set $S=\{a-mb\mid m\in \mathbb{Z} \text{ and } a-mb\geq 0\}$. This set $S$ naturally arises when you try to divide numbers. It shows up in the solution to the previous exercise.

Once you have this set $S$, either $0\in S$ or $0\not\in S$. If $0\in S$, what does that mean? If $0\not \in S$, can you show that $S$ is nonempty? What does the well ordering principle tell you?

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Page last modified on April 23, 2018, at 07:36 AM

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HomePage Schedule Pictures Writing Tips LaTeX Commands External Resources List of Problems List of Definitions List of Theorems List of Conjectures All Recent changes (use this to find a page that may have been lost)	Problem Existence Proof Of The Division Algorithm Please Login to access more options. Problem 37: (Existence Proof Of The Division Algorithm) Let $a$ and $b$ be integers with $b>0$. Prove that there exists integers $q$ and $r$ such that $a=qb+r$ and $0\leq r<b$. Click for a hint. Consider the set $S=\{a-mb\mid m\in \mathbb{Z} \text{ and } a-mb\geq 0\}$. This set $S$ naturally arises when you try to divide numbers. It shows up in the solution to the previous exercise. Once you have this set $S$, either $0\in S$ or $0\not\in S$. If $0\in S$, what does that mean? If $0\not \in S$, can you show that $S$ is nonempty? What does the well ordering principle tell you? The following pages link to this page. Problem.ExistenceProofOfTheDivisionAlgorithm Here are the old pages. Edit Page History Source Attach File Backlinks List Group Page last modified on April 23, 2018, at 07:36 AM
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