1,680 and 8,858 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.
1,680 = 24 × 3 × 5 × 7
1,680 is not a prime number, is a composite one.
8,858 = 2 × 43 × 103
8,858 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) (1,680; 8,858) = 2
Coprime numbers (prime to each other, relatively prime) (1,680; 8,858)? No.
The two numbers have common prime factors.
gcf (hcf, gcd) (1,680; 8,858) = 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:
8,858 ÷ 1,680 = 5 + 458
Step 2. Divide the smaller number by the above operation's remainder:
1,680 ÷ 458 = 3 + 306
Step 3. Divide the remainder of the step 1 by the remainder of the step 2:
458 ÷ 306 = 1 + 152
Step 4. Divide the remainder of the step 2 by the remainder of the step 3:
306 ÷ 152 = 2 + 2
Step 5. Divide the remainder of the step 3 by the remainder of the step 4:
152 ÷ 2 = 76 + 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) (1,680; 8,858) = 2
Coprime numbers (prime to each other, relatively prime) (1,680; 8,858)? No.
gcf (hcf, gcd) (1,680; 8,858) = 2