# 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$\gcd\!\left(n, k\right)$ Greatest common divisor
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)), Tuple(k, 1, m)), Product(Pow(PrimeNumber(k), Subscript(f, k)), Tuple(k, 1, m))), Product(Pow(PrimeNumber(k), Min(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)))))

## Topics using this entry

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

2019-06-18 07:49:59.356594 UTC