Fungrim entry: 6880d0

\gcd\!\left(a, b\right) = \max \left\{ d : d \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; d \mid a \;\mathbin{\operatorname{and}}\; d \mid b \right\}

Assumptions:

a \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; b \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; \left(a \ne 0 \;\mathbin{\operatorname{or}}\; b \ne 0\right)

TeX:

\gcd\!\left(a, b\right) = \max \left\{ d : d \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; d \mid a \;\mathbin{\operatorname{and}}\; d \mid b \right\}

a \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; b \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; \left(a \ne 0 \;\mathbin{\operatorname{or}}\; b \ne 0\right)

Definitions:

Fungrim symbol	Notation	Short description
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor
`Maximum`	$\mathop{\max}\limits_{x \in S} f(x)$	Maximum value of a set or function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("6880d0"),
    Formula(Equal(GCD(a, b), Maximum(Set(d, For(d), And(Element(d, ZZGreaterEqual(1)), Divides(d, a), Divides(d, b)))))),
    Variables(a, b),
    Assumptions(And(Element(a, ZZ), Element(b, ZZ), Or(NotEqual(a, 0), NotEqual(b, 0)))))

Topics using this entry

Greatest common divisor