# 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$\mathop{\max}\limits_{P\left(x\right)} f\!\left(x\right)$ 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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-21 11:44:15.926409 UTC