Question

Let * be the binary operation on N defined by a*b = HCF of a and b. Is * commutative? Is * associative? Does there exist identity for this binary operation on N?

Answer 1

The binary operation * on N is defined as a*b = HCF of a and b.  

It is known that HCF of a and b = HCF of b and a for a,b ∈ N.  

Therefore, a*b = b*a. Thus, the operation is commutative.

For a,b,c ∈ N , we have (a*b)*c = (HCF of a and b)*c = HCF of a,b and c  

a*(b*c) = a*(HCF of b and c) = HCF of a,b, and c  

Therefore, (a*b)*c = a*(b*c)  

Thus, the operation * is associative.  

Now, an element e ∈ N will be the identity for the operation if a*e = a = e*a, ∀ a ∈ N.  

But this relation is not true for any a ∈ N.  

Thus, the operation * does not have identity in N.

Answer 2

Answer:

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