# QuotientRemainder

QuotientRemainder[m,n]

gives a list of the quotient and remainder from division of m by n.

# Details

• QuotientRemainder is also known as the ratio and quantity "left over" from division of m by n.
• Integer mathematical function, suitable for both symbolic and numerical manipulation.
• QuotientRemainder[m,n] returns the ratio of m,n and the quantity "left over".

# Examples

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## Basic Examples(2)

Compute the quotient and remainder of two numbers:

Plot the sequence of quotients:

Plot the sequence of remainders:

## Scope(7)

QuotientRemainder works over integers:

Rational numbers:

Inexact real numbers:

Exact numbers:

Complex numbers:

Compute for large integers:

QuotientRemainder threads elementwise over lists:

## Applications(8)

### Basic Appications(3)

Plot the quotient of a number of division by 5:

Plot the remainder of a number of division by 5:

Plot the quotient of two integers:

### Number Theory(5)

Use NestWhileList to compute the quotient of positive arguments:

Compare with:

Use Floor to compute the quotient for integers:

Demonstrate how division works:

Count the number of positive integers less than 1000 divisible by 2 or 3, but not divisible by 6:

Direct count:

The Euclidean algorithm:

Compare with:

## Properties & Relations(2)

The first part of QuotientRemainder is the Quotient:

The second part of QuotientRemainder is the Mod:

## Neat Examples(2)

Plot the arguments of the Fourier transform of the QuotientRemainder:

Plot the Ulam spiral of the QuotientRemainder:

Wolfram Research (2007), QuotientRemainder, Wolfram Language function, https://reference.wolfram.com/language/ref/QuotientRemainder.html.

#### Text

Wolfram Research (2007), QuotientRemainder, Wolfram Language function, https://reference.wolfram.com/language/ref/QuotientRemainder.html.

#### CMS

Wolfram Language. 2007. "QuotientRemainder." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/QuotientRemainder.html.

#### APA

Wolfram Language. (2007). QuotientRemainder. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/QuotientRemainder.html

#### BibTeX

@misc{reference.wolfram_2024_quotientremainder, author="Wolfram Research", title="{QuotientRemainder}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/QuotientRemainder.html}", note=[Accessed: 14-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_quotientremainder, organization={Wolfram Research}, title={QuotientRemainder}, year={2007}, url={https://reference.wolfram.com/language/ref/QuotientRemainder.html}, note=[Accessed: 14-September-2024 ]}