### 420 and 126 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.

#### 420 = 2^{2} × 3 × 5 × 7;

420 is not a prime, is a composite number;

#### 126 = 2 × 3^{2} × 7;

126 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 (420; 126) = 2 × 3 × 7 = 42;

## Coprime numbers (relatively prime) (420; 126)? No.

The numbers have common prime factors.

gcf, hcf, gcd (126; 420) = 42.

### 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:

420 ÷ 126 = 3 + 42;

Step 2. Divide the smaller number by the above operation's remainder:

126 ÷ 42 = 3 + 0;

At this step, the remainder is zero, so we stop:

42 is the number we were looking for, the last remainder that is not zero.

This is the greatest common factor (divisor).

#### gcf, hcf, gcd (420; 126) = 42;

## Coprime numbers (relatively prime) (420; 126)? No.

gcf, hcf, gcd (126; 420) = 42.