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QuotientRemainder
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QuotientRemainder

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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

open allclose all

Basic Examples  (2)Summary of the most common use cases

Compute the quotient and remainder of two numbers:

In[1]:=1
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https://wolfram.com/xid/0bto17b3nqju-czivc
Out[1]=1

Plot the sequence of quotients:

In[1]:=1
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https://wolfram.com/xid/0bto17b3nqju-0nx3r
Out[1]=1

Plot the sequence of remainders:

In[2]:=2
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https://wolfram.com/xid/0bto17b3nqju-phkfa7
Out[2]=2

Scope  (7)Survey of the scope of standard use cases

QuotientRemainder works over integers:

In[2]:=2
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https://wolfram.com/xid/0bto17b3nqju-j8wraz
Out[2]=2

Rational numbers:

In[1]:=1
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https://wolfram.com/xid/0bto17b3nqju-79jea
Out[1]=1

Inexact real numbers:

In[1]:=1
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https://wolfram.com/xid/0bto17b3nqju-8t8al
Out[1]=1

Exact numbers:

In[1]:=1
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https://wolfram.com/xid/0bto17b3nqju-mvqzrl
Out[1]=1

Complex numbers:

In[1]:=1
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https://wolfram.com/xid/0bto17b3nqju-eyz8uq
Out[1]=1

Compute for large integers:

In[1]:=1
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https://wolfram.com/xid/0bto17b3nqju-cmsvct
Out[1]=1

QuotientRemainder threads elementwise over lists:

In[1]:=1
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https://wolfram.com/xid/0bto17b3nqju-bkrcck
Out[1]=1

Applications  (8)Sample problems that can be solved with this function

Basic Appications  (3)

Plot the quotient of a number of division by 5:

In[5]:=5
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https://wolfram.com/xid/0bto17b3nqju-csuwa9
Out[5]=5

Plot the remainder of a number of division by 5:

In[1]:=1
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https://wolfram.com/xid/0bto17b3nqju-7uyhs4
Out[1]=1

Plot the quotient of two integers:

In[1]:=1
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https://wolfram.com/xid/0bto17b3nqju-fpgpxl
Out[1]=1

Number Theory  (5)

Use NestWhileList to compute the quotient of positive arguments:

In[1]:=1
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https://wolfram.com/xid/0bto17b3nqju-t971f
In[2]:=2
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https://wolfram.com/xid/0bto17b3nqju-963m1
Out[2]=2

Compare with:

In[5]:=5
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https://wolfram.com/xid/0bto17b3nqju-pxyr3s
Out[5]=5

Use Floor to compute the quotient for integers:

In[1]:=1
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https://wolfram.com/xid/0bto17b3nqju-drekya
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bto17b3nqju-ngsz7w
Out[2]=2

Demonstrate how division works:

In[1]:=1
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https://wolfram.com/xid/0bto17b3nqju-oy5old
In[2]:=2
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https://wolfram.com/xid/0bto17b3nqju-752m7
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0bto17b3nqju-booqau
Out[1]=1

Direct count:

In[2]:=2
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https://wolfram.com/xid/0bto17b3nqju-byyp9v
Out[2]=2

The Euclidean algorithm:

In[1]:=1
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https://wolfram.com/xid/0bto17b3nqju-nacy6n
In[2]:=2
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https://wolfram.com/xid/0bto17b3nqju-f2bb9
Out[2]=2

Compare with:

In[3]:=3
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https://wolfram.com/xid/0bto17b3nqju-ii4c9n
Out[3]=3

Properties & Relations  (2)Properties of the function, and connections to other functions

The first part of QuotientRemainder is the Quotient:

In[1]:=1
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https://wolfram.com/xid/0bto17b3nqju-gnde2j
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bto17b3nqju-fc5gy
Out[2]=2

The second part of QuotientRemainder is the Mod:

In[1]:=1
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https://wolfram.com/xid/0bto17b3nqju-u8slch
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0bto17b3nqju-b4kib4
Out[2]=2

Neat Examples  (2)Surprising or curious use cases

Plot the arguments of the Fourier transform of the QuotientRemainder:

In[2]:=2
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https://wolfram.com/xid/0bto17b3nqju-1jmn54
Out[2]=2

Plot the Ulam spiral of the QuotientRemainder:

In[1]:=1
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https://wolfram.com/xid/0bto17b3nqju-zp3c6
In[2]:=2
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https://wolfram.com/xid/0bto17b3nqju-qhm0kv
Out[2]=2
Wolfram Research (2007), QuotientRemainder, Wolfram Language function, https://reference.wolfram.com/language/ref/QuotientRemainder.html.
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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.

✖
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.

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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

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

BibTeX

@misc{reference.wolfram_2025_quotientremainder, author="Wolfram Research", title="{QuotientRemainder}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/QuotientRemainder.html}", note=[Accessed: 26-March-2025 ]}

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@misc{reference.wolfram_2025_quotientremainder, author="Wolfram Research", title="{QuotientRemainder}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/QuotientRemainder.html}", note=[Accessed: 26-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_quotientremainder, organization={Wolfram Research}, title={QuotientRemainder}, year={2007}, url={https://reference.wolfram.com/language/ref/QuotientRemainder.html}, note=[Accessed: 26-March-2025 ]}

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@online{reference.wolfram_2025_quotientremainder, organization={Wolfram Research}, title={QuotientRemainder}, year={2007}, url={https://reference.wolfram.com/language/ref/QuotientRemainder.html}, note=[Accessed: 26-March-2025 ]}