Greatest Common Divisor of Integers/Examples/-5 and 5
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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:
- $\gcd \set {-5, 5} = 5$
$\blacksquare$