Are the Two Numbers 0 and 0 Coprime (Relatively Prime, Prime to Each Other)? Check if Their Greatest Common Factor, GCF, Is Equal to 1

Are the numbers 0 and 0 coprime (prime to each other, relatively prime)?

The numbers are not relatively prime... if:

If there is at least one number other than 1 that evenly divides the two numbers (without a remainder). Or...

Or, in other words, if their greatest (highest) common factor (divisor), gcf (hcf, gcd), is not equal to 1.

Calculate the greatest (highest) common factor (divisor),
gcf (hcf, gcd), of the two numbers

Zero is divisible by any number other than itself (the remainder is zero when dividing it by those numbers).

Zero has no greatest divisor since it has an infinite number of divisors. However, gcf (0; 0) is commonly defined as zero.


⇒ gcf (hcf, gcd) (0; 0) = 0 ≠ 1


Are the numbers 0 and 0 coprime (prime to each other, relatively prime)? No.

Are the two numbers coprime (relatively prime)?

Two natural numbers are coprime (relatively prime, prime to each other) - if there is no number that is evenly dividing both numbers (= without a remainder), that is, if their greatest (highest) common factor (divisor), gcf, or hcf, or gcd is 1.

Two natural numbers are not relatively prime - if there is at least one number that evenly divides the two numbers, that is, if their greatest common factor, gcf, is not 1.

The latest 10 pairs of numbers checked on whether they are coprime (prime to each other, relatively prime) or not

Coprime numbers (also called: numbers prime to each other, relatively prime, mutually prime)

  • The number "a" and "b" are said to be relatively prime, mutually prime, or coprime if the only positive integer that evenly divides both of them is 1.
  • The coprime numbers are pairs of (at least two) numbers that do not have any other common factor than 1.
  • When the only common factor is 1, then this is also equivalent to their greatest (highest) common factor (divisor) being 1.
  • Examples of pairs of coprime numbers:
  • The coprime numbers are not necessarily prime numbers themselves, for example 4 and 9 - these two numbers are not prime, they are composite numbers, since 4 = 2 × 2 = 22 and 9 = 3 × 3 = 32. But the gcf (4, 9) = 1, so they are coprime, or prime to each other, or relatively prime.
  • Sometimes, the coprime numbers in a pair are prime numbers themselves, for example (3 and 5), or (7 and 11), (13 and 23).
  • Some other times, the numbers that are prime to each other may or may not be prime, for example (5 and 6), (7 and 12), (15 and 23).
  • Examples of pairs of numbers that are not coprime:
  • 16 and 24 are not coprime, since they are both divisible by 1, 2, 4 and 8 (1, 2, 4 and 8 are their common factors).
  • 6 and 10 are not coprime, since they are both divisible by 1 and 2.
  • Some properties of the coprime numbers:
  • The greatest (highest) common factor (divisor) of two coprime numbers is always 1.
  • The least common multiple, LCM, of two coprimes is always their product: LCM (a, b) = a × b.
  • The numbers 1 and -1 are the only integers coprime to every integer, for example (1 and 2), (1 and 3), (1 and 4), (1 and 5), (1 and 6), and so on, are pairs of coprime numbers since their greatest (highest) common factor (divisor) is 1.
  • The numbers 1 and -1 are the only integers coprime to 0.
  • Any two prime numbers are always coprime, for example (2 and 3), (3 and 5), (5 and 7) and so on.
  • Any two consecutive numbers are co-prime, for example (1 and 2), (2 and 3), (3 and 4), (4 and 5), (5 and 6), (6 and 7), (7 and 8), (8 and 9), (9 and 10), and so on.
  • The sum of two coprime numbers, a + b, is always relatively prime to their product, a × b. For example, 7 and 10 are coprime numbers, 7 + 10 = 17 is relatively prime to 7 × 10 = 70. Another example, 9 and 11 are coprime, and their sum, 9 + 11 = 20 is relatively prime to their product, 9 × 11 = 99.
  • A quick way to determine whether two numbers are prime to each other is given by the Euclidean algorithm: The Euclidean Algorithm