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Fungrim entry: 25986e

gcd ⁣(k=1mpkek,k=1mpkfk)=k=1mpkmin(ek,fk)\gcd\!\left(\prod_{k=1}^{m} p_{k}^{{e}_{k}}, \prod_{k=1}^{m} p_{k}^{{f}_{k}}\right) = \prod_{k=1}^{m} p_{k}^{\min\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:
\gcd\!\left(\prod_{k=1}^{m} p_{k}^{{e}_{k}}, \prod_{k=1}^{m} p_{k}^{{f}_{k}}\right) = \prod_{k=1}^{m} p_{k}^{\min\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
GCDgcd ⁣(a,b)\gcd\!\left(a, b\right) Greatest common divisor
Productnf(n)\prod_{n} f(n) 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("25986e"),
    Formula(Equal(GCD(Product(Pow(PrimeNumber(k), Subscript(e, k)), For(k, 1, m)), Product(Pow(PrimeNumber(k), Subscript(f, k)), For(k, 1, m))), Product(Pow(PrimeNumber(k), Min(Subscript(e, k), Subscript(f, k))), For(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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2019-11-11 15:50:15.016492 UTC