# Fungrim entry: e922c4

$\gcd\!\left(a, b\right) = \min \left\{ a x + b y : x \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, y \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, a x + b y \ge 1 \right\}$
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) = \min \left\{ a x + b y : x \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, y \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, a x + b y \ge 1 \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
Minimum$\mathop{\min}\limits_{P\left(x\right)} f\!\left(x\right)$ Minimum value of a set or function
SetBuilder$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$ Set comprehension
ZZ$\mathbb{Z}$ Integers
Source code for this entry:
Entry(ID("e922c4"),
Formula(Equal(GCD(a, b), Minimum(SetBuilder(Add(Mul(a, x), Mul(b, y)), Tuple(x, y), And(Element(x, ZZ), Element(y, ZZ), GreaterEqual(Add(Mul(a, x), Mul(b, y)), 1)))))),
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