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Fungrim entry: 965ac0

{ax+by:xZandyZ}={nd:nZ}   where d=gcd ⁣(a,b)\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:aZandbZa \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{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
ZZZ\mathbb{Z} Integers
GCDgcd ⁣(a,b)\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))))

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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