Are 10 and 27 coprime (prime to each other, relatively prime)?
10 and 27 are coprime (relatively prime) - if there is no number other than 1 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.
10 = 2 × 5
10 is not a prime number, is a composite one.
27 = 33
27 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) (10; 27) = 1
Coprime numbers (prime to each other, relatively prime)
Coprime numbers (prime to each other, relatively prime) (10; 27)? Yes.
The numbers have no common prime factors.
gcf (hcf, gcd) (10; 27) = 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:
27 ÷ 10 = 2 + 7
Step 2. Divide the smaller number by the above operation's remainder:
10 ÷ 7 = 1 + 3
Step 3. Divide the remainder of the step 1 by the remainder of the step 2:
7 ÷ 3 = 2 + 1
Step 4. Divide the remainder of the step 2 by the remainder of the step 3:
3 ÷ 1 = 3 + 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) (10; 27) = 1
Coprime numbers (prime to each other, relatively prime) (10; 27)? Yes.
gcf (hcf, gcd) (10; 27) = 1
The final answer:
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10 and 27 are coprime (relatively prime) - if there is no number other than 1 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) (10; 27)? Yes.
gcf (hcf, gcd) (10; 27) = 1
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Are the two numbers coprime (relatively prime)?