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`	$\gcd\!\left(n, k\right)$	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

Greatest common divisor