# 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

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

2021-03-15 19:12:00.328586 UTC