Fungrim entry: 6880d0

\gcd\!\left(a, b\right) = \max\left(\left\{ d : d \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, d \mid a \,\mathbin{\operatorname{and}}\, d \mid b \right\}\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(\left\{ d : d \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, d \mid a \,\mathbin{\operatorname{and}}\, d \mid b \right\}\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`	$\gcd\!\left(n, k\right)$	Greatest common divisor
`Maximum`	$\mathop{\max}\limits_{P\left(x\right)} f\!\left(x\right)$	Maximum value of a set or function
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`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(SetBuilder(d, d, And(Element(d, ZZGreaterEqual(1)), Divides(d, a), Divides(d, b)))))),
    Variables(a, b),
    Assumptions(And(Element(a, ZZ), Element(b, ZZ), Or(Unequal(a, 0), Unequal(b, 0)))))

Topics using this entry

Greatest common divisor