Greatest Common Divisor of Integers/Examples/-8 and -36
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Example of Greatest Common Divisor of Integers
The greatest common divisor of $-8$ and $-36$ is:
- $\gcd \set {-8, -36} = 4$
Proof
The strictly positive divisors of $-8$ are:
- $\set {x \in \Z_{>0}: x \divides \paren {-8} } = \set {1, 2, 4, 8}$
The strictly positive divisors of $-36$ are:
- $\set {x \in \Z_{>0}: x \divides \paren {-36} } = \set {1, 2, 3, 4, 6, 9, 12, 18, 36}$
Of these, the common divisors are:
- $\set {x \in \Z_{>0}: x \divides \paren {-8} \land x \divides \paren {-36} } = \set {1, 2, 4}$
The greatest of these is $4$.
$\blacksquare$
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 $2 \text{-} 1$