Fungrim entry: 25986e

\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:

{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
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor
`Product`	$\prod_{n} f(n)$	Product
`Pow`	${a}^{b}$	Power
`PrimeNumber`	$p_{n}$	nth prime number
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	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)))))

Topics using this entry

Greatest common divisor