Relatively Prime Calculator

What Is a Relatively Prime Pair?

Two natural numbers are described as relatively prime (or coprime, mutually prime) when the only positive integer that divides both of them is 1. Equivalently, their greatest common divisor — often written as $\gcd(a,b)$ — equals 1. This condition is also captured by Bézout’s identity: there exist integers $x$ and $y$ such that $a x + b y = 1$. Because the numbers share no prime factor, any divisor of one cannot divide the other unless that divisor is 1.

A common misconception is that coprime numbers must themselves be primes. In fact, two composite numbers can be relatively prime as long as they have no overlapping prime factors. For instance, 14 and 15 are both composite ( $14 = 2 \times 7$, $15 = 3 \times 5$ ), yet they share only the factor 1, so they are coprime.

It follows directly that two even numbers can never be relatively prime, because every even number shares the factor 2. On the other hand, the number 1 is coprime with every integer, since its only divisor is 1 and therefore the GCD of 1 and any other number is always 1.

How to Check Whether Two Numbers Are Coprime

Identifying a relatively prime pair can be done through prime factorization:

1. Write each number as a product of its prime factors (using exponent notation when possible).
2. Compare the sets of primes that appear.
3. If the only common prime factor is none (i.e., the sets are disjoint), then the numbers are coprime; otherwise they are not.

Example 1 — 14 and 27
$14 = 2 \times 7$
$27 = 3^{3}$
No prime appears in both lists, so 14 and 27 are relatively prime.

Example 2 — 18 and 27
$18 = 2 \times 3^{2}$
$27 = 3^{3}$
The prime 3 appears in both factorizations, so 18 and 27 are not coprime. Their GCD is 9, not 1.

Generating a Number That Is Coprime to a Given Integer

To find a number that is relatively prime to a given integer $a$:

• First, compute the prime factorization of $a$.
• Any prime number that is not among those factors will be coprime to $a$.
• More generally, any natural number whose own prime factors avoid all the primes in $a$’s factorization will also be coprime to $a$.

Example: For $a = 45 = 3^{2} \times 5$, the prime 2 does not appear in its factorization, so 2 is relatively prime to 45. Also, a composite like $14 = 2 \times 7$ shares no prime factors with 45, confirming that 14 and 45 are coprime.

Coprimality in a Set of Numbers

When a set contains more than two integers, two levels of coprimality are distinguished:

• Setwise coprime — the entire set has a GCD of 1.
  Example: $\{4, 6, 21\}$ has GCD(4,6,21) = 1, so the set is setwise coprime.
• Pairwise coprime — every possible pair within the set is coprime.
  Example: $\{4, 7, 27\}$ is pairwise coprime, because (4,7), (4,27), and (7,27) are all coprime pairs.

Note that a set can be setwise coprime without being pairwise coprime (as $\{4, 6, 21\}$ shows: the pair (4,6) shares the factor 2).

Using the Online Free Relatively Prime Calculator

This free coprimes numbers checker online automates the factorization and comparison logic described above. Two modes are available:

• Pair mode — select “a pair of numbers,” enter the two integers, and the tool instantly reports whether they are relatively prime.
• Set mode — choose “a set of numbers” and enter up to ten integers. The calculator returns three possible outcomes: the set is setwise coprime, pairwise coprime, or not coprime at all.

With instant results and no manual steps, this relatively prime calculator provides a convenient way to check if numbers are relatively prime without performing factorisations yourself. Whether you are learning number theory or confirming properties of a set of integers, the coprime checker online offers a practical, time‑saving solution.