# HCF and LCM

• Nov 12th 2009, 10:32 AM
dangkhoa
HCF and LCM
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
• Nov 12th 2009, 02:01 PM
Jose27
Use your definitions, prove for example, that \$\displaystyle (x,y) \vert x\$ and \$\displaystyle (x,y) \vert y\$ and any other \$\displaystyle z\$ such that \$\displaystyle z \vert x\$ and \$\displaystyle z \vert y\$ we have \$\displaystyle z\vert (x,y)\$. Do the analogue for \$\displaystyle [x,y]\$. You should note however that \$\displaystyle (x,y)[x,y]\sim xy\$ (the equality doesn't necesarily hold) because you have to take into account the units \$\displaystyle u\$ and \$\displaystyle v\$.