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Theorem (Division Algorithm)

Let $a,n\in\mathbb{Z}$, with $n> 0$. Then there exists unique integers $q$ and $r$ (called the quotient and remainder) such that $a=qn+r$ where $0\leq r<n$.

Example

If we let $a=23$ and $n=5$, then we can write $23=4\cdot 5+3$. The quotient is $q=4$ and the remainder is $r=3$. Notice that $0\leq r<5$.

Non Example

Let $a=-23$ and $n=5$. We can write $-23=(-4)\cdot 5-3$, so it might be tempting to say that the quotient is $q=-4$ and the remainder is $r=-3$. However notice that it is not true that $0\leq r<5$. The remainder cannot be negative, and must be less than $n$. Instead we can write $-23=(-5)\cdot 5+2$. This means the quotient is $q=-5$ and the remainder is $r=2$.

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Page last modified on September 11, 2017, at 11:15 AM

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HomePage Current Problems All Problems Pictures ProblemsByWeek 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)	Example Division Algorithm Please Login to access more options. Theorem (Division Algorithm) Let $a,n\in\mathbb{Z}$, with $n> 0$. Then there exists unique integers $q$ and $r$ (called the quotient and remainder) such that $a=qn+r$ where $0\leq r<n$. Example If we let $a=23$ and $n=5$, then we can write $23=4\cdot 5+3$. The quotient is $q=4$ and the remainder is $r=3$. Notice that $0\leq r<5$. Non Example Let $a=-23$ and $n=5$. We can write $-23=(-4)\cdot 5-3$, so it might be tempting to say that the quotient is $q=-4$ and the remainder is $r=-3$. However notice that it is not true that $0\leq r<5$. The remainder cannot be negative, and must be less than $n$. Instead we can write $-23=(-5)\cdot 5+2$. This means the quotient is $q=-5$ and the remainder is $r=2$. Edit Page History Source Attach File Backlinks List Group Page last modified on September 11, 2017, at 11:15 AM
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