Coprime numbers, prime to each other, relatively prime: 1,023 and 6,196?

1,023 and 6,196: coprime numbers?

1,023 and 6,196 are coprime (relatively, mutually prime) if they have no common prime factors, that is, if their greatest (highest) common factor (divisor), gcf, hcf, gcd, is 1.

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

Approach 1. Integer numbers prime factorization:

Prime Factorization of a number: finding the prime numbers that multiply together to make that number.


1,023 = 3 × 11 × 31;
1,023 is not a prime, is a composite number;


6,196 = 22 × 1,549;
6,196 is not a prime, is a composite number;


Positive integers that are only dividing by themselves and 1 are called prime numbers. A prime number has only two factors: 1 and itself.


A composite number is a positive integer that has at least one factor (divisor) other than 1 and itself.


>> Integer numbers prime factorization


Calculate greatest (highest) common factor (divisor):

Multiply all the common prime factors, by the lowest exponents (if any).


But the two numbers have no common prime factors.


gcf, hcf, gcd (1,023; 6,196) = 1;
coprime numbers (relatively prime)



Coprime numbers (relatively prime) (1,023; 6,196)? Yes.
Numbers have no common prime factors.
gcf, hcf, gcd (1,023; 6,196) = 1.

Approach 2. Euclid's algorithm:

This algorithm involves the operation of dividing and calculating remainders.


'a' and 'b' are the two positive integers, 'a' >= 'b'.


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


If 'r' = 0, STOP. 'b' = the GCF (HCF, GCD) of 'a' and 'b'.


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



Step 1. Divide the larger number by the smaller one:
6,196 ÷ 1,023 = 6 + 58;
Step 2. Divide the smaller number by the above operation's remainder:
1,023 ÷ 58 = 17 + 37;
Step 3. Divide the remainder from the step 1 by the remainder from the step 2:
58 ÷ 37 = 1 + 21;
Step 4. Divide the remainder from the step 2 by the remainder from the step 3:
37 ÷ 21 = 1 + 16;
Step 5. Divide the remainder from the step 3 by the remainder from the step 4:
21 ÷ 16 = 1 + 5;
Step 6. Divide the remainder from the step 4 by the remainder from the step 5:
16 ÷ 5 = 3 + 1;
Step 7. Divide the remainder from the step 5 by the remainder from the step 6:
5 ÷ 1 = 5 + 0;
At this step, the remainder is zero, so we stop:
1 is the number we were looking for, the last remainder that is not zero.
This is the greatest common factor (divisor).


gcf, hcf, gcd (1,023; 6,196) = 1;


>> Euclid's algorithm

Coprime numbers (relatively prime) (1,023; 6,196)? Yes.
gcf, hcf, gcd (1,023; 6,196) = 1.

Final answer:

1,023 and 6,196 are coprime (relatively, mutually prime) if they have no common prime factors, that is, if their greatest (highest) common factor (divisor), gcf, hcf, gcd, is 1.
Coprime numbers (relatively prime) (1,023; 6,196)? Yes.
gcf, hcf, gcd (1,023; 6,196) = 1.

More operations of this kind:

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Coprime numbers (numbers prime to each other, relatively prime, mutually prime)

Integers "a" and "b" are said to be relatively prime, mutually prime, or coprime if the only positive integer that divides both of them is 1. This is equivalent to their only common positive factor being 1. This is also equivalent to their greatest common factor (divisor) being 1.

For example, 16 and 17 are coprime, being commonly divisible by only 1, but 16 and 24 are not, because they are both divisible by 8. The numbers 1 and -1 are the only integers coprime to every integer, and they are the only integers to be coprime with 0. A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm: Euclid's algorithm


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