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Find identity element
Given:
\( a * b = \mathrm{HCF}(a,b), \quad a,b \in \mathbb{N} \)
Identity Definition:
\( a * e = a \Rightarrow \mathrm{HCF}(a,e) = a \)
Analysis:
This implies:
\( a \mid e \quad \text{for all } a \in \mathbb{N} \)
So, \( e \) must be divisible by every natural number.
But no such finite natural number exists.
Conclusion:
❌ Identity element does NOT exist in \( \mathbb{N} \)