Greatest Common Divisor of Integers/Examples/-5 and 5/Proof 1
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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
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 $5$.
$\blacksquare$
Sources
- 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $2$: Divisibility Theory in the Integers: $2.2$ The Greatest Common Divisor: Example $2 \text{-} 1$