Greatest Common Divisor of Integers/Examples/-5 and 5

Example of Greatest Common Divisor of Integers

The greatest common divisor of $-5$ and $5$ is:

$\gcd \set {-5, 5} = 5$

Proof 1

The strictly positive divisors of $-5$ are:

$\set {x \in \Z_{>0}: x \divides \paren {-5} } = \set {1, 5}$

The strictly positive divisors of $5$ are:

$\set {x \in \Z_{>0}: x \divides 5 } = \set {1, 5}$

Of these, the common divisors are:

$\set {x \in \Z_{>0}: x \divides \paren {-5} \land x \divides 5 } = \set {1, 5}$

The greatest of these is $5$.

$\blacksquare$

Proof 2

From GCD of Integer and its Negative:

$\gcd \set {-5, 5} = 5$

$\blacksquare$