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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 (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.
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