456 and 2,049 are not coprime (relatively, mutually prime) if they have common prime factors, that is, if their greatest (highest) common factor (divisor), gcf, hcf, gcd, is not 1.
Calculate the greatest (highest) common factor (divisor), gcf, hcf, gcd
Approach 1. Integer numbers prime factorization:
Prime Factorization of a number: finding the prime numbers that multiply together to make that number.
456 = 23 × 3 × 19;
456 is not a prime, is a composite number;
2,049 = 3 × 683;
2,049 is not a prime, is a composite number;
Positive integers that are only dividing by themselves and 1 are called prime numbers. A prime number has only two factors: 1 and itself.
A composite number is a positive integer that has at least one factor (divisor) other than 1 and itself.
Calculate greatest (highest) common factor (divisor):
Multiply all the common prime factors, by the lowest exponents (if any).
gcf, hcf, gcd (456; 2,049) = 3;
Coprime numbers (relatively prime) (456; 2,049)? No.
The numbers have common prime factors.
gcf, hcf, gcd (456; 2,049) = 3.
Approach 2. Euclid's algorithm:
This algorithm involves the operation of dividing and calculating remainders.
'a' and 'b' are the two positive integers, 'a' >= 'b'.
Divide 'a' by 'b' and get the remainder, 'r'.
If 'r' = 0, STOP. 'b' = the GCF (HCF, GCD) of 'a' and 'b'.
Else: Replace ('a' by 'b') & ('b' by 'r'). Return to the division step above.
Step 1. Divide the larger number by the smaller one:
2,049 ÷ 456 = 4 + 225;
Step 2. Divide the smaller number by the above operation's remainder:
456 ÷ 225 = 2 + 6;
Step 3. Divide the remainder from the step 1 by the remainder from the step 2:
225 ÷ 6 = 37 + 3;
Step 4. Divide the remainder from the step 2 by the remainder from the step 3:
6 ÷ 3 = 2 + 0;
At this step, the remainder is zero, so we stop:
3 is the number we were looking for, the last remainder that is not zero.
This is the greatest common factor (divisor).
gcf, hcf, gcd (456; 2,049) = 3;
Coprime numbers (relatively prime) (456; 2,049)? No.
gcf, hcf, gcd (456; 2,049) = 3.