GCD of Integer and its Negative
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Theorem
Let $a \in \Z$ be an integer.
Then:
- $\gcd \set {a, -a} = \size a$
where:
- $\gcd$ denotes greatest common divisor
- $\size a$ denotes the absolute value of $a$.
Proof
From Integer Divisor Results, the divisors of $a$ include $a$ itself.
From Integer Divides its Negative, $a \divides \paren {-a}$.
Thus we have:
- $a \divides a$
and:
- $a \divides -a$
and so:
- $\gcd \set {a, -a} \ge \size a$
From Absolute Value of Integer is not less than Divisors, there is no divisor of $a$ which is greater than $a$.
That is:
- $\gcd \set {a, -a} \le \size a$
Hence the result.
$\blacksquare$