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Thread: HCF and LCM

  1. #1
    Newbie dangkhoa's Avatar
    Nov 2009

    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
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  2. #2
    Super Member
    Apr 2009
    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 $\displaystyle v$.
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