Greatest Common Divisor of Integers/Examples/-12 and 30
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Example of Greatest Common Divisor of Integers
The greatest common divisor of $-12$ and $30$ is:
- $\gcd \set {-12, 30} = 6$
Proof
The strictly positive divisors of $-12$ are:
- $\set {x \in \Z_{>0}: x \divides \paren {-12} } = \set {1, 2, 3, 4, 6, 12}$
The strictly positive divisors of $30$ are:
- $\set {x \in \Z_{>0}: x \divides 30} = \set {1, 2, 3, 5, 6, 10, 15, 30}$
Of these, the common divisors are:
- $\set {x \in \Z_{>0}: x \divides \paren {-12} \land x \divides 30} = \set {1, 2, 3, 6}$
The greatest of these is $6$.
$\blacksquare$
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 $2 \text{-} 1$