Greatest Common Divisor of Integers/Examples/12 and -8

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 $\text {2-5}$