Natural number LCM


THEOREM   

One is a left identity element of natural number LCM.

$$\forall a \in \mathbb{N} : \lcm(1, a) = a$$

Proof available

THEOREM   

One is a right identity element of natural number LCM.

$$\forall a \in \mathbb{N} : \lcm(a, 1) = a$$

Proof available

THEOREM   

Natural number LCM is idempotent.

$$\forall a \in \mathbb{N} : \lcm(a, a) = a$$

Proof available

THEOREM   

Natural number LCM is commutative.

$$\forall a,b \in \mathbb{N} : \lcm(a, b) = \lcm(b, a)$$

Proof available

THEOREM   

Natural number LCM is associative.

$$\forall a,b,c \in \mathbb{N} : \lcm(\lcm(a, b), c) = \lcm(a, \lcm(b, c))$$

Proof available

THEOREM   

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

Proof available

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