  In this paper we included some basic properties for weighted greatest common
divisors, and discuss how to speed up computing the weighted greatest common
divisor. By ordering the 'weights' we are able to significantly shorten the
operations to computing wgcd. In the absence of an efficient algorithm for
computing wgcd by ordering the weights, and using $\gcd$, we significantly
reduce the numbers for which we want to compute wgcd. As a final result in this
paper we prove that: If $\mathbf{x} = (x_{0},\dots ,x_{n})\in
\mathbb{Z}^{n+1}$, with weights $\mathfrak{w}=(q_{0},\dots ,q_{n})$ and
$q_{0}\leq \cdots \leq q_{n}$, then ${\rm wgcd}_w(\mathbf{x}) = {\rm
wgcd}_w(y_0,y_1,\dots,y_n)$, where $y_i = \gcd(x_i,\dots,x_n)$, and $y_0\leq
y_1 \leq\dots \leq y_n$.

General Mathematics