Modular Arithmetic: quotient and remainder

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Daniel Mansfield

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Daniel Mansfield 7 years, 8 months ago

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Name	Status	Author	Last Modified
Modular Arithmetic: quotient and remainder	Ready to use	Daniel Mansfield	11/03/2025 06:29
Copy of Modular Arithmetic: quotient and remainder	draft	Paul Emanuel	28/06/2018 05:37
Modular Arithmetic: quotient and remainder	Ready to use	Alexander Corner	06/09/2021 10:21

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Question statement

Given integers $a,b \in \mathbb Z$ we can write $a$ in the form

$a = qb + r$

for some $q,r \in \mathbb Z$ where $0 \leq r < b$ . The numbers $q$ and $r$ are called the quotient and remainder when $a$ is divided by $b$ :

$\displaystyle \frac{a}{b} = q + \frac{r}{b}$ .

In modular arithmetic (in this case, modulo $b$ ) our interest is focused on the remainder. The operation $a \operatorname{mod} b = r$ .

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			Name	Type	Generated Value

p	integer	6
r	integer	4
p2	integer	4
r2	integer	1

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Which numbers have a remainder of $\var{r}$ when divided by $\var{p}$ ? These numbers would be considered equal in modulo $\var{p}$ .

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$\var{r+p}$

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$\var{r-p}$

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$\var{r+p+1}$

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$\var{r-p-1}$

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$\var{r-2p+1}$

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$\var{r-2p}$

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$\var{r+3p}$

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Advice

The first two questions introduce the idea that different numbers in $\mathbb Z$ can be the same in modulo $b$ . For example in modulo $\var{p}$ the numbers

$\var{p}, \var{2p}, \var{3p}, \cdots$

are equivalent to zero. While the numbers

$\var{p2+1}, \var{2p2+1}, \var{3p2+1}, \cdots$

are equivalent to $1 \pmod{\var{p2}}$ .

The next question introduce a simple divisibility test: $a=qb = qb+0$ exactly when $a\equiv 0\pmod{b}$ . This is just another way to say that $b$ divides $a$ , or that $a$ is a multiple of $b$ . In particular

$\var{p*p2} = \var{p}\times \var{p2}$ is divisible by $\var{p}$ and $\var{p2}$ .

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Daniel Mansfield 7 years, 8 months ago

Daniel Mansfield 7 years, 8 months ago

Daniel Mansfield 7 years, 8 months ago

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Modular Arithmetic: quotient and remainder

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Daniel Mansfield 7 years, 8 months ago

Daniel Mansfield 7 years, 8 months ago

Daniel Mansfield 7 years, 8 months ago

Error in variable testing condition

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Generated value: integer

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Choose several from a list

Advanced settings

Generated value: `integer`