# Fungrim entry: 965ac0

$\left\{ a x + b y : x \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, y \in \mathbb{Z} \right\} = \left\{ n d : n \in \mathbb{Z} \right\}\; \text{ where } d = \gcd\!\left(a, b\right)$
Assumptions:$a \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, b \in \mathbb{Z}$
TeX:
\left\{ a x + b y : x \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, y \in \mathbb{Z} \right\} = \left\{ n d : n \in \mathbb{Z} \right\}\; \text{ where } d = \gcd\!\left(a, b\right)

a \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, b \in \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
SetBuilder$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$ Set comprehension
ZZ$\mathbb{Z}$ Integers
GCD$\gcd\!\left(a, b\right)$ Greatest common divisor
Source code for this entry:
Entry(ID("965ac0"),
Formula(Where(Equal(SetBuilder(Add(Mul(a, x), Mul(b, y)), Tuple(x, y), And(Element(x, ZZ), Element(y, ZZ))), SetBuilder(Mul(n, d), n, Element(n, ZZ))), Equal(d, GCD(a, b)))),
Variables(a, b),
Assumptions(And(Element(a, ZZ), Element(b, ZZ))))

## Topics using this entry

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

2019-08-19 14:38:23.809000 UTC