# Natural number GCD

THEOREM

One is a left zero element of natural number GCD.

$$\forall a \in \mathbb{N} : \gcd(1, a) = 1$$
Proof.
THEOREM

One is a right zero element of natural number GCD.

$$\forall a \in \mathbb{N} : \gcd(a, 1) = 1$$
Proof.
THEOREM

Natural number GCD is idempotent.

$$\forall x \in \mathbb{N} : \gcd(x, x) = x$$
Proof.
THEOREM

Natural number GCD is commutative.

$$\forall x,y \in \mathbb{N} : \gcd(x, y) = \gcd(y, x)$$
Proof.
THEOREM

Natural number GCD is associative.

$$\forall x,y \in \mathbb{N} : \gcd(\gcd(x, y), z) = \gcd(x, \gcd(y, z))$$
Proof.
THEOREM

$$(\mathbb{N}, \gcd)$$ is an Abelian semigroup.

Proof.

Natural number GCD is closed, associative, and commutative.