Are the Two Numbers 17 and 85 Coprime (Relatively Prime, Prime to Each Other)? Check if Their Greatest Common Factor, GCF, Is Equal to 1

Are the numbers 17 and 85 coprime (prime to each other, relatively prime)?

17 and 85 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

Method 1. The divisibility of numbers:

Divide the larger number by the smaller one.

When dividing the two numbers, there is no remainder:

85 ÷ 17 = 5 + 0

⇒ 85 = 17 × 5

⇒ 85 is divisible by 17

⇒ 17 is a factor (a divisor) of 85


Consequently, gcf (hcf, gcd) (17; 85) = 17 ≠ 1


Coprime numbers (prime to each other, relatively prime) (17; 85)? No.
gcf (hcf, gcd) (17; 85) = 17 ≠ 1
Scroll down for the 2nd method...

Method 2. The prime factorization:

The prime factorization of a number: finding the prime numbers that multiply together to make that number.


17 is a prime number, it cannot be prime factorized.

85 = 5 × 17
85 is not a prime number, is a composite one.


Prime number: a number that is divisible (dividing evenly) only by 1 and itself. A prime number has only two factors: 1 and itself.

Composite number: a natural number that has at least one other factor than 1 and itself.

» Check whether a number is prime or not. The prime factorization of composite numbers



Calculate the greatest (highest) common factor (divisor), gcf (hcf, gcd):

Multiply all the common prime factors of the two numbers, taken by their smallest exponents (powers).

gcf (hcf, gcd) (17; 85) = 17 ≠ 1



Coprime numbers (prime to each other, relatively prime) (17; 85)? No.
85 has all the prime factors of the number 17.
gcf (hcf, gcd) (17; 85) = 17 ≠ 1

Other similar operations with coprime numbers:

» Are the two numbers 10 and 82 coprime (relatively prime, prime to each other)?

» Are the two numbers 19 and 94 coprime (relatively prime, prime to each other)?


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.

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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