Use your definitions, prove for example, that and and any other such that and we have . Do the analogue for . You should note however that (the equality doesn't necesarily hold) because you have to take into account the units and .

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- November 12th 2009, 10:32 AM #1
## 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

- November 12th 2009, 02:01 PM #2

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