The Lcm(a, b, ...) calling sequence computes the unique least common right multiple of the given ordinal numbers. It returns either an ordinal data structure, a nonnegative integer, or a polynomial with positive integer coefficients.

If $m$ is the largest ordinal among all the arguments, the least common right multiple equals either $m·c$ for a positive integer $c$ or, $m·\mathrm{\omega }$.

If some of the arguments are parametric ordinals and the least common right multiple cannot be determined, an error is raised.

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

$a≔\mathrm{Ordinal}\left(\left[\left[1\,1\right]\,\left[0\,1\right]\right]\right)$

$a≔\mathbf{\omega }\+1$

$b≔\mathrm{Ordinal}\left(\left[\left[2\,2\right]\,\left[1\,2\right]\,\left[0\,1\right]\right]\right)$

$b≔{\mathbf{\omega }}^2\cdot 2\+\mathbf{\omega }\cdot 2\+1$

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

$c≔{\mathbf{\omega }}^2\cdot 3\+\mathbf{\omega }\cdot 2\+1$

$\mathrm{l1}≔\mathrm{Lcm}\left(a\,b\,c\right)$

$\mathrm{l1}≔{\mathbf{\omega }}^2\cdot 6\+\mathbf{\omega }\cdot 2\+1$

$\mathrm{Div}\left(\mathrm{l1}\,a\right)$

$\mathbf{\omega }\cdot 6\+2\,0$

$\mathrm{Div}\left(\mathrm{l1}\,b\right)$

$3,0$

$\mathrm{Div}\left(\mathrm{l1}\,c\right)$

$2,0$

$\mathrm{l2}≔\mathrm{Lcm}\left(a+1\,b\,c\right)$

$\mathrm{l2}≔{\mathbf{\omega }}^3$

$\mathrm{Div}\left(\mathrm{l2}\,a+1\right)$

${\mathbf{\omega }}^2\,0$

$\mathrm{Div}\left(\mathrm{l2}\,b\right)$

$\mathbf{\omega }\,0$

$\mathrm{Div}\left(\mathrm{l2}\,c\right)$

$\mathbf{\omega }\,0$

$\mathrm{Lcm}\left(a\,b\,c\,0\right)$

$0$

$\mathrm{Lcm}\left(a+1\,2\right)$

$\mathbf{\omega }\+2$

$\mathrm{Lcm}\left(a+1\,3\right)$

${\mathbf{\omega }}^2$

$d≔\mathrm{Ordinal}\left(\left[\left[2\,x\right]\,\left[1\,2\right]\,\left[0\,1\right]\right]\right)$

$d≔{\mathbf{\omega }}^2\cdot x\+\mathbf{\omega }\cdot 2\+1$

$\mathrm{Lcm}\left(a\,b\,d\right)$

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

$\mathrm{Lcm}\left(a\,b\,\mathrm{Eval}\left(d\,x=x+1\right)\right)$

Error, (in Ordinals:-Lcm) unable to compute lcm

$\mathrm{Lcm}\left(a\,b\,\mathrm{Eval}\left(d\,x=2x+2\right)\right)$

${\mathbf{\omega }}^2\cdot \left(2x+2\right)\+\mathbf{\omega }\cdot 2\+1$

$\mathrm{Div}\left(\,b\right)$

$x+1,0$

$\mathrm{Div}\left(\,a\right)$

$\mathbf{\omega }\cdot \left(2x+2\right)\+2\,0$

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

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