6,598 and 5,054 are not relatively prime -- if there is at least one number that evenly divides the two numbers (without a remainder) -- or, in other words -- if their greatest (highest) common factor (divisor), gcf (hcf, gcd), is not 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.
6,598 = 2 × 3,299
6,598 is not a prime number, is a composite one.
5,054 = 2 × 7 × 192
5,054 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).
gcf (hcf, gcd) (6,598; 5,054) = 2
Coprime numbers (prime to each other, relatively prime) (6,598; 5,054)? No.
The two numbers have common prime factors.
gcf (hcf, gcd) (5,054; 6,598) = 2
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:
6,598 ÷ 5,054 = 1 + 1,544
Step 2. Divide the smaller number by the above operation's remainder:
5,054 ÷ 1,544 = 3 + 422
Step 3. Divide the remainder of the step 1 by the remainder of the step 2:
1,544 ÷ 422 = 3 + 278
Step 4. Divide the remainder of the step 2 by the remainder of the step 3:
422 ÷ 278 = 1 + 144
Step 5. Divide the remainder of the step 3 by the remainder of the step 4:
278 ÷ 144 = 1 + 134
Step 6. Divide the remainder of the step 4 by the remainder of the step 5:
144 ÷ 134 = 1 + 10
Step 7. Divide the remainder of the step 5 by the remainder of the step 6:
134 ÷ 10 = 13 + 4
Step 8. Divide the remainder of the step 6 by the remainder of the step 7:
10 ÷ 4 = 2 + 2
Step 9. Divide the remainder of the step 7 by the remainder of the step 8:
4 ÷ 2 = 2 + 0
At this step, the remainder is zero, so we stop:
2 is the number we were looking for - the last non-zero remainder.
This is the greatest (highest) common factor (divisor).
gcf (hcf, gcd) (6,598; 5,054) = 2
Coprime numbers (prime to each other, relatively prime) (6,598; 5,054)? No.
gcf (hcf, gcd) (5,054; 6,598) = 2