# Relatively Prime Numbers and Polynomials

Two numbers are said to be
**
relatively prime
**
if their
greatest common factor
(
GCF
) is
$1$
.

**
Example 1:
**

The factors of $20$ are $1,2,4,5,10,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}20$ .

The factors of $33$ are $1,3,11,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}33$ .

The only common factor is $1$ . So, the GCF is $1$ .

Therefore, $20\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}33$ are relatively prime.

**
Example 2:
**

The factors of $45$ are $1,3,5,9,15,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}45$ .

The factors of $51$ are $1,3,17,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}51$ .

The greatest common factor here is $3$ .

Therefore,
$45\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}51$
are
**
not
**
relatively prime.

The definition can be extended to polynomials . In this case, there should be no common variable or polynomial factors, and the scalar coefficients should have a GCF of $1$ .

**
Example 3:
**

The polynomial $3{x}^{2}+21x+18$ can be factored as

$3{x}^{2}+21x+18=3(x+1)(x+6)$ .

The polynomial $5x+10$ can be factored as

$5x+10=5(x+2)$ .

$3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}5$ are relatively prime, and none of the binomial factors are shared. So, the two polynomials

$3{x}^{2}+21x+18\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}5x+10$

are relatively prime.

**
Example 4:
**

The polynomial ${x}^{2}-3x-4$ can be factored as

${x}^{2}-3x-4=(x+1)(x-4)$ .

The polynomial $3{x}^{2}+21x+18$ can be factored as

$3{x}^{2}+21x+18=3(x+1)(x+6)$ .

The two polynomials share a binomial factor:

$(x+1)$
.

So

${x}^{2}-3x-4\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}3{x}^{2}+21x+18$

are
**
not
**
relatively prime.