Are the Two Numbers 111 and 95,987 Coprime (Relatively Prime, Prime to Each Other)? Check if Their Greatest Common Factor, GCF, Is Equal to 1
Are the numbers 111 and 95,987 coprime (prime to each other, relatively prime)?
111 and 95,987 are coprime (relatively prime)... if:
If there is no number other than 1 that evenly divides (without a remainder) both numbers. Or...
Or, in other words, if their greatest (highest) common factor (divisor), gcf (hcf, gcd), is equal to 1.
Calculate the greatest (highest) common factor (divisor),
gcf (hcf, gcd), of the two numbers
Method 1. The prime factorization:
The prime factorization of a number: finding the prime numbers that multiply together to make that number.
111 = 3 × 37
111 is not a prime number, is a composite one.
95,987 is a prime number, it cannot be prime factorized.
Prime number: a number that is divisible (dividing evenly) only by 1 and itself. A prime number has only two factors: 1 and itself.
Composite number: 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) (111; 95,987) = 1
Coprime numbers (prime to each other, relatively prime)
Coprime numbers (prime to each other, relatively prime) (111; 95,987)? Yes.
The numbers have no common prime factors.
gcf (hcf, gcd) (111; 95,987) = 1
Scroll down for the 2nd method...
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:
95,987 ÷ 111 = 864 + 83
Step 2. Divide the smaller number by the above operation's remainder:
111 ÷ 83 = 1 + 28
Step 3. Divide the remainder of the step 1 by the remainder of the step 2:
83 ÷ 28 = 2 + 27
Step 4. Divide the remainder of the step 2 by the remainder of the step 3:
28 ÷ 27 = 1 + 1
Step 5. Divide the remainder of the step 3 by the remainder of the step 4:
27 ÷ 1 = 27 + 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) (111; 95,987) = 1
Coprime numbers (prime to each other, relatively prime) (111; 95,987)? Yes.
gcf (hcf, gcd) (111; 95,987) = 1
Other similar operations with coprime numbers:
Are the two numbers coprime (relatively prime)?
Two natural numbers are coprime (relatively prime, prime to each other) - if there is no number that is evenly dividing both numbers (= without a remainder), that is, if their greatest (highest) common factor (divisor), gcf, or hcf, or gcd is 1.
Two natural numbers are not relatively prime - if there is at least one number that evenly divides the two numbers, that is, if their greatest common factor, gcf, is not 1.