Coprime integers¶

In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1.

Version: v1 · 2026-09-28 · History
Domain-specific #: 8724
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Number Theory → Mathematics

Core Idea¶

Coprime integers is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common.

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Numbers That Share Nothing

Two numbers are coprime if the only number that splits both of them into equal groups is 1. For example, 8 and 9: you can't split both into groups of 2, or 3, or any other same number. But 6 and 9 aren't coprime, because both split into groups of 3.

Numbers With Only 1 in Common

Two whole numbers are coprime, also called relatively prime, when the only positive number that divides evenly into both of them is 1. For example, 8 and 9 are coprime: 8 is divisible by 1, 2, 4 and 8, and 9 by 1, 3 and 9, and the only one they share is 1. Neither 8 nor 9 is a prime number, but they are still coprime to each other. On the other hand, 6 and 9 are not coprime, because 3 divides both.

Relatively Prime Pair

Two integers are coprime (also called relatively prime or mutually prime) if the only positive integer that divides both is 1. Equivalently, their greatest common divisor (GCD) is 1. This also means no prime number divides both of them. Being coprime is a relationship between two numbers, not a property of each one alone: 8 and 9 are coprime even though neither is prime, while 6 and 9 are not, since both are divisible by 3.

In number theory, two integers a and b are coprime, relatively prime, or mutually prime if the only positive integer dividing both is 1, equivalently if gcd(a, b) = 1. A direct consequence is that any prime dividing a does not divide b, and vice versa, so the two numbers share no prime factors. One also says a is prime to b, or coprime with b. Coprimality is a relation on pairs, not a property of either number individually: 8 and 9 are coprime although neither is prime, while 6 and 9 are not, because 3 divides both. The concept depends only on shared divisors, not on the size or primality of the numbers involved.

Scope of Application¶

Notation and testing. In their 1989 textbook Concrete Mathematics, Ronald Graham, Donald Knuth, and Oren Patashnik proposed an alternative notation a\perp b to indicate that and are relatively prime and that the term.
Notation and testing. The number of integers coprime with a positive integer , between 1 and , is given by Euler's totient function, also known as Euler's phi function, .
Probability of coprimality. Here refers to the Riemann zeta function, the identity relating the product over primes to is an example of an Euler product, and the evaluation of as is the Basel problem.
Notation and testing. When the integers and are coprime, the standard way of expressing this fact in mathematical notation is to indicate that their greatest common divisor is one, by the formula or .
Notation and testing. A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm and its faster variants such as binary GCD algorithm or Lehmer's GCD algorithm.

Clarity¶

A clear use of Coprime integers names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1.

Manages Complexity¶

Coprime integers compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—a fast way to determine whether two numbers are coprime is given by the Euclidean algorithm and its faster variants such as binary GCD algorithm or Lehmer's GCD algorithm.—and the practical consequence—hence the probability that two numbers are both divisible by is and the probability that at least one of.

Abstract Reasoning¶

Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
State the relation. Use the source-grounded identity: In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1.
Check operation and conditions. The number of integers coprime with a positive integer , between 1 and , is given by Euler's totient function, also known as Euler's phi function, .
Demand recognition evidence.

Knowledge Transfer¶

Within the home domain. Knowledge about Coprime integers transfers literally when a new case preserves the same carrier type, relation, and recognition test. In their 1989 textbook Concrete Mathematics, Ronald Graham, Donald Knuth, and Oren Patashnik proposed an alternative notation a\perp b to indicate that and are relatively prime and that the term "prime" be used instead of coprime (as in is prime to ).

Neighborhood in Abstraction Space¶

Coprime integers sits in a moderately populated region (45^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Number-Theoretic Properties & Tests (20 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08