The Gcd(a, b, ...) calling sequence computes the unique greatest common left divisor of the given ordinal numbers. It returns either an ordinal data structure, a nonnegative integer, or a polynomial with positive integer coefficients.

If some of the arguments are parametric ordinals and the greatest common left divisor cannot be determined, an error is raised.

$\mathrm{with}\left(\mathrm{Ordinals}\right)$

$\left[\mathrm{`+`}\&comma;\mathrm{`.`}\&comma;\mathrm{`<`}\&comma;\mathrm{<=}\&comma;\mathrm{Add}\&comma;\mathrm{Base}\&comma;\mathrm{Dec}\&comma;\mathrm{Decompose}\&comma;\mathrm{Div}\&comma;\mathrm{Eval}\&comma;\mathrm{Factor}\&comma;\mathrm{Gcd}\&comma;\mathrm{Lcm}\&comma;\mathrm{LessThan}\&comma;\mathrm{Log}\&comma;\mathrm{Max}\&comma;\mathrm{Min}\&comma;\mathrm{Mult}\&comma;\mathrm{Ordinal}\&comma;\mathrm{Power}\&comma;\mathrm{Split}\&comma;\mathrm{Sub}\&comma;\mathrm{`^`}\&comma;\mathrm{degree}\&comma;\mathrm{lcoeff}\&comma;\log \&comma;\mathrm{lterm}\&comma;\mathrm{\omega }\&comma;\mathrm{quo}\&comma;\mathrm{rem}\&comma;\mathrm{tcoeff}\&comma;\mathrm{tdegree}\&comma;\mathrm{tterm}\right]$

$a≔\mathrm{Ordinal}\left(\left[\left[\mathrm{\omega }\&comma;1\right]\&comma;\left[1\&comma;2\right]\&comma;\left[0\&comma;1\right]\right]\right)$

$a≔{\mathbf{\omega }}^{\mathbf{\omega }}\&plus;\mathbf{\omega }\cdot 2\&plus;1$

$b≔\mathrm{Ordinal}\left(\left[\left[3\&comma;1\right]\&comma;\left[1\&comma;1\right]\&comma;\left[0\&comma;1\right]\right]\right)$

$b≔{\mathbf{\omega }}^3\&plus;\mathbf{\omega }\&plus;1$

$c≔\mathrm{Ordinal}\left(\left[\left[2\&comma;1\right]\&comma;\left[1\&comma;3\right]\&comma;\left[0\&comma;1\right]\right]\right)$

$c≔{\mathbf{\omega }}^2\&plus;\mathbf{\omega }\cdot 3\&plus;1$

$\mathrm{Gcd}\left(a\&comma;b\&comma;c\right)$

$\mathbf{\omega }\&plus;1$

$\mathrm{Div}\left(a\&comma;\right)$

${\mathbf{\omega }}^{\mathbf{\omega }}\&plus;2\&comma;0$

$\mathrm{Div}\left(b\&comma;\right)$

${\mathbf{\omega }}^2\&plus;1\&comma;0$

$\mathrm{Div}\left(c\&comma;\right)$

$\mathbf{\omega }\&plus;3\&comma;0$

$\mathrm{Gcd}\left(12\&comma;20\&comma;30\right)$

$2$

$\mathrm{Gcd}\left(18\&comma;12·b\&comma;30·c\right)$

$6$

$\mathrm{Gcd}\left(3\&comma;\mathrm{\omega }\right)$

$3$

$\mathrm{Gcd}\left(3\&comma;\mathrm{\omega }\&comma;\mathrm{\omega }+1\right)$

$1$

$d≔\mathrm{Ordinal}\left(\left[\left[2\&comma;x\right]\&comma;\left[1\&comma;3\right]\&comma;\left[0\&comma;1\right]\right]\right)$

$d≔{\mathbf{\omega }}^2\cdot x\&plus;\mathbf{\omega }\cdot 3\&plus;1$

$\mathrm{Gcd}\left(a\&comma;b\&comma;d\right)$

$\mathbf{\omega }\&plus;1$

$e≔\mathrm{Ordinal}\left(\left[\left[2\&comma;1\right]\&comma;\left[1\&comma;1\right]\&comma;\left[0\&comma;1\right]\right]\right)$

$e≔{\mathbf{\omega }}^2\&plus;\mathbf{\omega }\&plus;1$

$\mathrm{Gcd}\left(d\&comma;e\right)$

$\mathbf{\omega }\&plus;1$

$\mathrm{Div}\left(d\&comma;\right)$

$\mathbf{\omega }\cdot x\&plus;3\&comma;0$

$\mathrm{Div}\left(e\&comma;\right)$

$\mathbf{\omega }\&plus;1\&comma;0$

$f≔\mathrm{Ordinal}\left(\left[\left[3\&comma;1\right]\&comma;\left[1\&comma;3\right]\&comma;\left[0\&comma;1\right]\right]\right)$

$f≔{\mathbf{\omega }}^3\&plus;\mathbf{\omega }\cdot 3\&plus;1$

$\mathrm{Gcd}\left(d\&comma;f\right)$

Error, (in Ordinals:-Gcd) cannot determine if x is nonzero

$\mathrm{Gcd}\left(\mathrm{Eval}\left(d\&comma;x=x+1\right)\&comma;f\right)$

$\mathbf{\omega }\cdot 3\&plus;1$

$g≔\mathrm{Ordinal}\left(\left[\left[4\&comma;1\right]\&comma;\left[2\&comma;x+1\right]\right]\right)$

$g≔{\mathbf{\omega }}^4\&plus;{\mathbf{\omega }}^2\cdot \left(x+1\right)$

$h≔\mathrm{Ordinal}\left(\left[\left[3\&comma;2\right]\&comma;\left[1\&comma;y+1\right]\&comma;\left[0\&comma;z\right]\right]\right)$

$h≔{\mathbf{\omega }}^3\cdot 2\&plus;\mathbf{\omega }\cdot \left(y+1\right)\&plus;z$

$\mathrm{Gcd}\left(g\&comma;h\right)$

$\mathbf{\omega }\cdot \left(y+1\right)\&plus;z$

$\mathrm{Div}\left(g\&comma;\right)$

${\mathbf{\omega }}^3\&plus;\mathbf{\omega }\cdot \left(x+1\right)\&comma;0$

$\mathrm{Div}\left(h\&comma;\right)$

${\mathbf{\omega }}^2\cdot 2\&plus;1\&comma;0$

$\mathrm{Gcd}\left(4\&comma;h\&comma;\mathrm{\omega }+6\right)$

$\mathrm{igcd}\left(2\&comma;z\right)$

The Ordinals[Gcd] command was introduced in Maple 2015.

For more information on Maple 2015 changes, see Updates in Maple 2015.