Greatest Common Divisor of Integers/Examples/12 and -8
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Example of Greatest Common Divisor of Integers
The greatest common divisor of $12$ and $-8$ is:
- $\gcd \set {12, -8} = 4$
Proof
The strictly positive divisors of $12$ are:
- $\set {x \in \Z_{>0}: x \divides 12} = \set {1, 2, 3, 4, 6, 12}$
The strictly positive divisors of $-8$ are:
- $\set {x \in \Z_{>0}: x \divides \paren {-8} } = \set {1, 2, 4, 8}$
Of these, the common divisors are:
- $\set {x \in \Z_{>0}: x \divides 12 \land x \divides \paren {-8} } = \set {1, 2, 4}$
The greatest of these is $4$.
$\blacksquare$
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-2}$ Divisibility: Example $\text {2-5}$