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CoprimeQ[n1,n2]

yields True if n1 and n2 are relatively prime, and yields False otherwise.

CoprimeQ[n1,n2,…]

yields True if all pairs of the ni are relatively prime, and yields False otherwise.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
GaussianIntegers  
Applications  
Basic Applications  
Number Theory  
Properties & Relations  
Neat Examples  
See Also
Tech Notes
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CoprimeQ

CoprimeQ[n1,n2]

yields True if n1 and n2 are relatively prime, and yields False otherwise.

CoprimeQ[n1,n2,…]

yields True if all pairs of the ni are relatively prime, and yields False otherwise.

Details

  • CoprimeQ is typically used to test whether two numbers are relatively prime.
  • Integers are relatively prime if their greatest common divisor is 1.
  • CoprimeQ[n1,n2] returns False unless n1,n2 are manifestly relatively prime.
  • With the setting GaussianIntegers->True, CoprimeQ tests whether Gaussian integers are relatively prime.
  • CoprimeQ works over Gaussian integers.

Examples

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Basic Examples  (2)

Test whether two numbers are relatively prime:

The numbers and are not relatively prime:

Scope  (4)

CoprimeQ works over integers:

Gaussian integers:

Test for large integers:

CoprimeQ threads elementwise over lists:

Options  (1)

GaussianIntegers  (1)

Test whether is composite over integers:

Gaussian integers:

Applications  (8)

Basic Applications  (3)

Highlight numbers that are coprime to :

Generate random numbers coprime to a given number:

Plot random pairs of coprime numbers:

Visualize when two numbers are coprime:

Number Theory  (5)

Use CoprimeQ to compute Euler's totient function:

Use CoprimeQ to check for trivial GCDs:

Find the fraction of pairs of the first numbers that are relatively prime:

The result is close to :

Compute the modular inverse of coprime numbers:

Use ExtendedGCD:

Database encryption and decryption:

Key generation:

Encrypted data:

Decrypted data:

Properties & Relations  (9)

Coprime numbers have a greatest common divisor GCD equal to :

The least common multiple LCM of two coprime numbers is equal to their product:

The number of divisors of a number preserves multiplication for coprime numbers:

Coprime numbers a and b satisfy for some integers x and y:

The numbers and are the only integers coprime to every integer:

Prime numbers are relatively prime to each other:

EulerPhi gives the count of the positive integers up to n that are relatively prime to n:

Coprime numbers a and n satisfy :

If a and b are coprime, then so are any powers and :

Neat Examples  (2)

Plot numbers that are coprime:

Plot the Ulam spiral of a sequence of coprime numbers:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2025_coprimeq, author="Wolfram Research", title="{CoprimeQ}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/CoprimeQ.html}", note=[Accessed: 09-August-2025]}

BibLaTeX

@online{reference.wolfram_2025_coprimeq, organization={Wolfram Research}, title={CoprimeQ}, year={2007}, url={https://reference.wolfram.com/language/ref/CoprimeQ.html}, note=[Accessed: 09-August-2025]}