HCF and LCM

dangkhoa · #1 $Old$ November 12th, 2009, 11:32 AM

Suppose that x, y in R*\U(R) that is x,y are non-zero, non-unit in R, where R is UFD. Let x = u.{p_1^(a_1)....p_t^(a_t)}
y = v{p_1^(b_1)....p_t^(b_t)}
where a_i and b_i >= 0 , 1=<i=<t and u,v are in U(R).

Let c_i =max{a_i , b_i} and d_i = min{a_i , b_i}.

Show that HCF{x,y} and LCM{x,y} exist and
HCF{x,y} = (x,y) = p_1^(d_1).....p_t^(d_t)
LCM{x,y} = [x,y] = p_1^(c_1).....p_t^(c_t)

Deduce that (x,y)[x,y] = xy

How do you approach to this problem

Thank you very much

Jose27 · #2 $Old$ November 12th, 2009, 03:01 PM

Use your definitions, prove for example, that $(x,y) \vert x$ and $(x,y) \vert y$ and any other $z$ such that $z \vert x$ and $z \vert y$ we have $z\vert (x,y)$ . Do the analogue for $[x,y]$ . You should note however that $(x,y)[x,y]\sim xy$ (the equality doesn't necesarily hold) because you have to take into account the units $u$ and $v$ .