On the set N of all natural numbers, define the operation * on N by min=gcd (m, n) for all m,ne N. Show that is commutative as well as associative.

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soution (i) Commutativity For all ( m, n in N, ) we have ( operatorname{gcd}(m, n)=operatorname{gcd}(n, m) ) [ therefore m * n=n+m forall m, n in N ] Hence, ( * ) is commutative on ( N ). (ii) Associativity Let ( m, n, p in N . ) Then, [ begin{aligned} (m * n) * p &=[operatorname{gcd}(m, n)]^{*} p &=operatorname{gcd}[g operatorname{cd}{(m, n), p}] &=operatorname{gcd}[m, operatorname{gcd}(n, p)}] end{aligned} ] ( [because text { gcd of three numbers }=operatorname{gcd}{(g c d text { of any two, th }) ) [ =operatorname{gcd}(m, n * p)=m *(n * p) ] Hence, ( ^{*} ) is associative on ( N ).

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