6,259 and 9,891 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.
6,259 = 11 × 569
6,259 is not a prime number, is a composite one.
9,891 = 32 × 7 × 157
9,891 is not a prime number, is a composite one.
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.
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) (6,259; 9,891) = 1
Coprime numbers (prime to each other, relatively prime)
Coprime numbers (prime to each other, relatively prime) (6,259; 9,891)? Yes.
The numbers have no common prime factors.
gcf (hcf, gcd) (6,259; 9,891) = 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:
9,891 ÷ 6,259 = 1 + 3,632
Step 2. Divide the smaller number by the above operation's remainder:
6,259 ÷ 3,632 = 1 + 2,627
Step 3. Divide the remainder of the step 1 by the remainder of the step 2:
3,632 ÷ 2,627 = 1 + 1,005
Step 4. Divide the remainder of the step 2 by the remainder of the step 3:
2,627 ÷ 1,005 = 2 + 617
Step 5. Divide the remainder of the step 3 by the remainder of the step 4:
1,005 ÷ 617 = 1 + 388
Step 6. Divide the remainder of the step 4 by the remainder of the step 5:
617 ÷ 388 = 1 + 229
Step 7. Divide the remainder of the step 5 by the remainder of the step 6:
388 ÷ 229 = 1 + 159
Step 8. Divide the remainder of the step 6 by the remainder of the step 7:
229 ÷ 159 = 1 + 70
Step 9. Divide the remainder of the step 7 by the remainder of the step 8:
159 ÷ 70 = 2 + 19
Step 10. Divide the remainder of the step 8 by the remainder of the step 9:
70 ÷ 19 = 3 + 13
Step 11. Divide the remainder of the step 9 by the remainder of the step 10:
19 ÷ 13 = 1 + 6
Step 12. Divide the remainder of the step 10 by the remainder of the step 11:
13 ÷ 6 = 2 + 1
Step 13. Divide the remainder of the step 11 by the remainder of the step 12:
6 ÷ 1 = 6 + 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) (6,259; 9,891) = 1
Coprime numbers (prime to each other, relatively prime) (6,259; 9,891)? Yes.
gcf (hcf, gcd) (6,259; 9,891) = 1