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On the set N of all natural numbers define the ooperation * on N by m*n = gcd (m,n) for all m,n in N Showat * is commutative as well as associative |
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Answer» Solution :(i) commutativity for all m,n in N we have gcd (m,n) =gcd(n.m) Therefore m*n=n*m,n in N (ii) Associativity LET m,n p in N Then (m*n)*p=[gcd(m,n,p}] =gcd[{gcd{(m,n),p}] [`therefore` gcd of three NUMBERS =gcd {(gcd of any t wo THIRD )}] =gcd (m,n*p)=m*(n*p) Hence * is associatie on N |
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