Fungrim entry: 1bbdaf

\operatorname{lcm}\!\left(a + n b, b\right) = \frac{\left|a + n b\right| \operatorname{lcm}\!\left(a, b\right)}{\left|a\right|}

Assumptions:

a \in \mathbb{Z} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, b \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

TeX:

\operatorname{lcm}\!\left(a + n b, b\right) = \frac{\left|a + n b\right| \operatorname{lcm}\!\left(a, b\right)}{\left|a\right|}

a \in \mathbb{Z} \setminus \left\{0\right\} \,\mathbin{\operatorname{and}}\, b \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`LCM`	$\operatorname{lcm}\!\left(a, b\right)$	Least common multiple
`Abs`	$\left\|z\right\|$	Absolute value
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("1bbdaf"),
    Formula(Equal(LCM(Add(a, Mul(n, b)), b), Div(Mul(Abs(Add(a, Mul(n, b))), LCM(a, b)), Abs(a)))),
    Variables(a, b, n),
    Assumptions(And(Element(a, SetMinus(ZZ, Set(0))), Element(b, ZZ), Element(n, ZZ))))

Topics using this entry

Greatest common divisor