IsCoprime

Tests whether the arguments are pairwise relatively prime.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageCoprimeQ✓↗

IsCoprime(a, b, …)true when every pair of arguments is coprime

Typed signatures

Type	Package	Applies
(number+) -> boolean	number-theory	always

Domain: Number theory

Details

• Pairwise, not collectively: $\gcd$ of the whole set being 1 is not enough — every pair must itself be coprime.
• Extends to Gaussian integers, over the same GCD this widens.
• Wolfram's CoprimeQ.

Examples

Basic #

live ↗

In$\mathrm{IsCoprime}(4, 9)$Out$\mathrm{True}$

Both are divisible by 3

live ↗

In$\mathrm{IsCoprime}(6, 9)$Out$\mathrm{False}$

Several arguments: every pair must be coprime

live ↗

In$\mathrm{IsCoprime}(2, 3, 5)$Out$\mathrm{True}$

No shared factor, so coprime

live ↗

In$\mathrm{IsCoprime}(8, 11)$Out$\mathrm{True}$

Both even, so not coprime

live ↗

In$\mathrm{IsCoprime}(2, 4)$Out$\mathrm{False}$

Scope #

Gaussian integers: $2+i$ and $2-i$ are non-associate primes

live ↗

In$\mathrm{IsCoprime}(2+i, 2-i)$Out$\mathrm{True}$

Listable

live ↗

In$\mathrm{IsCoprime}(3, \bigl\lbrack4, 6\bigr\rbrack)$Out$\bigl\lbrack\mathrm{True}, \mathrm{False}\bigr\rbrack$

Several arguments must be pairwise coprime; signs are ignored

live ↗

In$\mathrm{IsCoprime}(2, 3, -5, 7)$Out$\mathrm{True}$

Gaussian integers: both divisible by $1+i$ up to a unit

live ↗

In$\mathrm{IsCoprime}(5+i, 1-i)$Out$\mathrm{False}$

Large integers handled exactly; these share a factor

live ↗

In$\mathrm{IsCoprime}(2^{100}-1, 3^{100}-1)$Out$\mathrm{False}$

Threads over a list against 6

live ↗

In$\mathrm{IsCoprime}(\bigl\lbrack1, 2, 3, 4, 5\bigr\rbrack, 6)$Out$\bigl\lbrack\mathrm{True}, \mathrm{False}, \mathrm{False}, \mathrm{False}, \mathrm{True}\bigr\rbrack$

Distinct primes are coprime

live ↗

In$\mathrm{IsCoprime}(2, 3)$Out$\mathrm{True}$

Applications #

Consecutive integers are always coprime

live ↗

In$\mathrm{IsCoprime}(5, 6)$Out$\mathrm{True}$

Two distinct odd primes share no factor

live ↗

In$\mathrm{IsCoprime}(5, 3)$Out$\mathrm{True}$

Properties #

$2^3$ and $3^2$: coprime powers of distinct primes

live ↗

In$\mathrm{IsCoprime}(8, 9)$Out$\mathrm{True}$

A prime is coprime to anything it does not divide

live ↗

In$\mathrm{IsCoprime}(6, 11)$Out$\mathrm{True}$

No shared factor between 5 and $2^2\cdot 3$

live ↗

In$\mathrm{IsCoprime}(5, 12)$Out$\mathrm{True}$

Symbolic arguments are taken as not provably coprime

live ↗

In$\mathrm{IsCoprime}(a, b)$Out$\mathrm{False}$

Distinct primes are pairwise coprime

live ↗

In$\mathrm{IsCoprime}(\mathrm{NthPrime}(10), \mathrm{NthPrime}(15), \mathrm{NthPrime}(20), \mathrm{NthPrime}(25))$Out$\mathrm{True}$

No shared factor between 7 and $2\cdot 5$

live ↗

In$\mathrm{IsCoprime}(7, 10)$Out$\mathrm{True}$

Possible issues #

Pairwise, not collectively: $\gcd(6,10,15)=1$ yet no pair is coprime

live ↗

In$\mathrm{IsCoprime}(6, 10, 15)$Out$\mathrm{False}$

About

Examples	21
Checked against	Wolfram Language 21 agree
Crosswalk	Wolfram Language

See also: GCD, IsPrime