Are the Two Numbers 40 and 21 Relatively Prime (Coprime, Prime to Each Other)? Online Calculator

Are the numbers 40 and 21 coprime (prime to each other, relatively prime)? The relationship to their greatest common factor

40 and 21 are coprime (relatively prime)... if:

If there is no number other than 1 that evenly divides (without a remainder) both numbers. Or...

Or, in other words, if their greatest (highest) common factor (divisor), gcf (hcf, gcd), is equal to 1.

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

Method 1. The prime factorization:

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


40 = 23 × 5
40 is not a prime number, is a composite one.

21 = 3 × 7
21 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 (decompose numbers into prime factors)



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).

But the numbers have no common prime factors.

gcf (hcf, gcd) (40; 21) = 1
Coprime numbers (prime to each other, relatively prime)



Coprime numbers (prime to each other, relatively prime) (40; 21)? Yes.
The numbers have no common prime factors.
gcf (hcf, gcd) (21; 40) = 1
Scroll down for the 2nd method...

Method 2. The Euclidean Algorithm:

This algorithm involves the process of dividing numbers and calculating the remainders.

'a' and 'b' are the two natural numbers, 'a' >= 'b'.

Divide 'a' by 'b' and get the remainder of the operation, 'r'.

If 'r' = 0, STOP. 'b' = the gcf (hcf, gcd) of 'a' and 'b'.

Else: Replace ('a' by 'b') and ('b' by 'r'). Return to the step above.

» The Euclidean Algorithm



Step 1. Divide the larger number by the smaller one:
40 ÷ 21 = 1 + 19
Step 2. Divide the smaller number by the above operation's remainder:
21 ÷ 19 = 1 + 2
Step 3. Divide the remainder of the step 1 by the remainder of the step 2:
19 ÷ 2 = 9 + 1
Step 4. Divide the remainder of the step 2 by the remainder of the step 3:
2 ÷ 1 = 2 + 0
At this step, the remainder is zero, so we stop:
1 is the number we were looking for - the last non-zero remainder.
This is the greatest (highest) common factor (divisor).

gcf (hcf, gcd) (40; 21) = 1


Coprime numbers (prime to each other, relatively prime) (40; 21)? Yes.
gcf (hcf, gcd) (21; 40) = 1


Other similar operations with coprime numbers:

» Are the two numbers 50 and 58 coprime (relatively prime, prime to each other)?

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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