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Question
Let * be the binary operation on N defined by a * b = HCF of a and b.
Does there exist identity for this binary operation one N ?
Solution
Let e be the identity element. Then,
\[a * e = a = e * a, \forall a \in N\]
\[HCF\left( a, e \right) = a = HCF\left( e, a \right), \forall a \in N\]
\[ \Rightarrow HCF\left( a, e \right) = a, \forall a \in N\]
We cannot find e that satisfies this condition.
So, the identity element with respect to * does not exist in N.
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