maxim comú divisor álgebra

mcd{x,y} = x [M] y (intersecció en divisors)
mcm{x,y} = x [W] y (reunió en divisors)


mcd{(1/x),(1/y)} = (1/x) [M] (1/y)
mcm{(1/x),(1/y)} = (1/x) [W] (1/y)


mcd{2,6} = 2 [M] 6 = 2
mcm{2,6} = 2 [W] 6 = 6


mcd{(1/2),(1/6)} = (1/2) [M] (1/6) = (1/6)
mcm{(1/2),(1/6)} = (1/2) [W] (1/6) = (1/2)


2·3=6
(1/6)·3=(1/2)


mcd{mcd{x,y},z} =  mcd{x,mcd{y,z}}
mcm{mcm{x,y},z} =  mcm{x,mcm{y,z}}


mcd{mcd{x,y},z} =  mcd{x,mcd{y,z}}
mcd{mcd{x,y},z} = mcd{x,y} [M] z
mcd{mcd{x,y},z} = (x [M] y) [M] z
mcd{mcd{x,y},z} = x [M] (y [M] z)
mcd{mcd{x,y},z} = x [M] mcd{y,z}
mcd{mcd{x,y},z} =  mcd{x,mcd{y,z}}


mcm{mcm{x,y},z} =  mcm{x,mcm{y,z}}
mcm{mcm{x,y},z} = mcm{x,y} [W] z
mcm{mcm{x,y},z} = (x [W] y) [W] z
mcm{mcm{x,y},z} = x [W] (y [W] z)
mcm{mcm{x,y},z} = x [W] mcm{y,z}
mcm{mcm{x,y},z} =  mcm{x,mcm{y,z}}


mcd{x,y} =  mcd{y,x}
mcm{x,y} =  mcm{y,x}


mcd{x,x} =  x
mcm{x,x} =  x


mcm{mcd{x,y},z}=mcd{mcm{x,z},mcm{y,z}}
mcd{mcm{x,y},z}=mcm{mcd{x,z},mcd{y,z}}


m·k=n <==> m | n <==> mcd{m,n}=m <==> mcm{m,n}=n
(1/n)·k=(1/m) <==> (1/n) | (1/m) <==> mcm{(1/m),(1/n)}=(1/m) <==> mcd{(1/m),(1/n)}=(1/n)