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Fungrim entry: 6cefd7

lcm ⁣(k=1mpkek,k=1mpkfk)=k=1mpkmax(ek,fk)\operatorname{lcm}\!\left(\prod_{k=1}^{m} {p_{k}}^{{e}_{k}}, \prod_{k=1}^{m} {p_{k}}^{{f}_{k}}\right) = \prod_{k=1}^{m} {p_{k}}^{\max\left({e}_{k}, {f}_{k}\right)}
Assumptions:ekZ0andfkZ0andmZ0{e}_{k} \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, {f}_{k} \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, m \in \mathbb{Z}_{\ge 0}
TeX:
\operatorname{lcm}\!\left(\prod_{k=1}^{m} {p_{k}}^{{e}_{k}}, \prod_{k=1}^{m} {p_{k}}^{{f}_{k}}\right) = \prod_{k=1}^{m} {p_{k}}^{\max\left({e}_{k}, {f}_{k}\right)}

{e}_{k} \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, {f}_{k} \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, m \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
LCMlcm ⁣(a,b)\operatorname{lcm}\!\left(a, b\right) Least common multiple
Productnf ⁣(n)\prod_{n} f\!\left(n\right) Product
Powab{a}^{b} Power
PrimeNumberpnp_{n} nth prime number
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("6cefd7"),
    Formula(Equal(LCM(Product(Pow(PrimeNumber(k), Subscript(e, k)), Tuple(k, 1, m)), Product(Pow(PrimeNumber(k), Subscript(f, k)), Tuple(k, 1, m))), Product(Pow(PrimeNumber(k), Max(Subscript(e, k), Subscript(f, k))), Tuple(k, 1, m)))),
    Variables(e, f, m),
    Assumptions(And(Element(Subscript(e, k), ZZGreaterEqual(0)), Element(Subscript(f, k), ZZGreaterEqual(0)), Element(m, ZZGreaterEqual(0)))))

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

2019-08-17 11:32:46.829430 UTC