# Highest common factor

The highest common factor (hcf) of a set of integers is the largest integer which is a factor of all of them (in other words, which divides all of them).

For example, the hcf of $18$, $21$ and $30$ is $3$.

The highest common factor is also known as the greatest common divisor (gcd).

A highest common factor of a set of polynomials is a polynomial of greatest possible degree which divides all of them.

Examples:

• An hcf of $3x^2$ and $2x$ is $x$; note that $-x$ is also an hcf of these.
• An hcf of $x^2+2x$ and $x^2+3x+2$ is $x+2$.

If the polynomials have integer coefficients, we usually require the hcf to also have the greatest possible coefficients (in magnitude), so an hcf of $6x^2$ and $4xy$ is $2x$; $-2x$ is also an hcf, but $x$ is not an hcf as the coefficient $1$ is smaller than $2$.