Are the two numbers 1,953 and 9,719 coprime (relatively prime, prime to each other)? Check if their greatest common factor, gcf, is equal to 1

Are 1,953 and 9,719 coprime (prime to each other, relatively prime)?

1,953 and 9,719 are coprime (relatively prime) -- if there is no number that evenly divides the both numbers (without a remainder) -- that is -- if their greatest (highest) common factor (divisor), gcf (hcf, gcd), is 1.

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

Method 1. The prime factorization:

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


1,953 = 32 × 7 × 31
1,953 is not a prime number, is a composite one.


9,719 is a prime number, it cannot be prime factorized.


The numbers that are only divisible by 1 and themselves are called prime numbers. A prime number has only two factors: 1 and itself.


A composite number is a natural number that has at least one other factor than 1 and itself.


>> The prime factorization of 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).


But the numbers have no common prime factors.


gcf (hcf, gcd) (1,953; 9,719) = 1
Coprime numbers (prime to each other, relatively prime)



Coprime numbers (prime to each other, relatively prime) (1,953; 9,719)? Yes.
The numbers have no common prime factors.
gcf (hcf, gcd) (1,953; 9,719) = 1

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.



Step 1. Divide the larger number by the smaller one:
9,719 ÷ 1,953 = 4 + 1,907
Step 2. Divide the smaller number by the above operation's remainder:
1,953 ÷ 1,907 = 1 + 46
Step 3. Divide the remainder of the step 1 by the remainder of the step 2:
1,907 ÷ 46 = 41 + 21
Step 4. Divide the remainder of the step 2 by the remainder of the step 3:
46 ÷ 21 = 2 + 4
Step 5. Divide the remainder of the step 3 by the remainder of the step 4:
21 ÷ 4 = 5 + 1
Step 6. Divide the remainder of the step 4 by the remainder of the step 5:
4 ÷ 1 = 4 + 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) (1,953; 9,719) = 1


>> The Euclidean Algorithm

Coprime numbers (prime to each other, relatively prime) (1,953; 9,719)? Yes.
gcf (hcf, gcd) (1,953; 9,719) = 1

The final answer:

1,953 and 9,719 are coprime (relatively prime) -- if there is no number that evenly divides the both numbers (without a remainder) -- that is -- if their greatest (highest) common factor (divisor), gcf (hcf, gcd), is 1.
Coprime numbers (prime to each other, relatively prime) (1,953; 9,719)? Yes.
gcf (hcf, gcd) (1,953; 9,719) = 1

More operations of this kind:

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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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