Lcm - Maple Help

Ordinals

 Lcm
 least common right multiple of ordinals

 Calling Sequence Lcm(a, b, ...)

Parameters

 a, b, ... - ordinals, nonnegative integers, or polynomials with positive integer coefficients

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Ordinals}\right):$
 > $a≔\mathrm{Ordinal}\left(\left[\left[1,1\right],\left[0,1\right]\right]\right)$
 ${a}{≔}{\mathbf{\omega }}{+}{1}$ (1)
 > $b≔\mathrm{Ordinal}\left(\left[\left[2,2\right],\left[1,2\right],\left[0,1\right]\right]\right)$
 ${b}{≔}{{\mathbf{\omega }}}^{{2}}{\cdot }{2}{+}{\mathbf{\omega }}{\cdot }{2}{+}{1}$ (2)
 > $c≔\mathrm{Ordinal}\left(\left[\left[2,3\right],\left[1,2\right],\left[0,1\right]\right]\right)$
 ${c}{≔}{{\mathbf{\omega }}}^{{2}}{\cdot }{3}{+}{\mathbf{\omega }}{\cdot }{2}{+}{1}$ (3)
 > $\mathrm{l1}≔\mathrm{Lcm}\left(a,b,c\right)$
 ${\mathrm{l1}}{≔}{{\mathbf{\omega }}}^{{2}}{\cdot }{6}{+}{\mathbf{\omega }}{\cdot }{2}{+}{1}$ (4)
 > $\mathrm{Div}\left(\mathrm{l1},a\right)$
 ${\mathbf{\omega }}{\cdot }{6}{+}{2}{,}{0}$ (5)
 > $\mathrm{Div}\left(\mathrm{l1},b\right)$
 ${3}{,}{0}$ (6)
 > $\mathrm{Div}\left(\mathrm{l1},c\right)$
 ${2}{,}{0}$ (7)
 > $\mathrm{l2}≔\mathrm{Lcm}\left(a+1,b,c\right)$
 ${\mathrm{l2}}{≔}{{\mathbf{\omega }}}^{{3}}$ (8)
 > $\mathrm{Div}\left(\mathrm{l2},a+1\right)$
 ${{\mathbf{\omega }}}^{{2}}{,}{0}$ (9)
 > $\mathrm{Div}\left(\mathrm{l2},b\right)$
 ${\mathbf{\omega }}{,}{0}$ (10)
 > $\mathrm{Div}\left(\mathrm{l2},c\right)$
 ${\mathbf{\omega }}{,}{0}$ (11)

Any of the arguments can be a nonnegative integer.

 > $\mathrm{Lcm}\left(a,b,c,0\right)$
 ${0}$ (12)
 > $\mathrm{Lcm}\left(a+1,2\right)$
 ${\mathbf{\omega }}{+}{2}$ (13)
 > $\mathrm{Lcm}\left(a+1,3\right)$
 ${{\mathbf{\omega }}}^{{2}}$ (14)

Parametric examples.

 > $d≔\mathrm{Ordinal}\left(\left[\left[2,x\right],\left[1,2\right],\left[0,1\right]\right]\right)$
 ${d}{≔}{{\mathbf{\omega }}}^{{2}}{\cdot }{x}{+}{\mathbf{\omega }}{\cdot }{2}{+}{1}$ (15)
 > $\mathrm{Lcm}\left(a,b,d\right)$
 > $\mathrm{Lcm}\left(a,b,\mathrm{Eval}\left(d,x=x+1\right)\right)$
 > $\mathrm{Lcm}\left(a,b,\mathrm{Eval}\left(d,x=2x+2\right)\right)$
 ${{\mathbf{\omega }}}^{{2}}{\cdot }\left({2}{}{x}{+}{2}\right){+}{\mathbf{\omega }}{\cdot }{2}{+}{1}$ (16)
 > $\mathrm{Div}\left(,b\right)$
 ${x}{+}{1}{,}{0}$ (17)
 > $\mathrm{Div}\left(,a\right)$
 ${\mathbf{\omega }}{\cdot }\left({2}{}{x}{+}{2}\right){+}{2}{,}{0}$ (18)

Compatibility

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