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Let * be the binary operation on N def...

Let * be the binary operation on `N` defined by a*b=`H C F` of `a` and `b` . Does there exist identity for this binary operation on `N` ?

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Let a, b `in` N
`therefore ` a* b = H.C.F of a and b
= H.C.F of b and a
= b *a
`therefore ` Operation * is commutative.
Let a,b, `in` N
`therefore` a* (b*c) = a* H.C.F of a and b
= H.C.F of a, b and c
= (H.C.F of a and b) * c
`therefore` Operation * is associative
Let e `in` N be the identity element then
a * e = e* a = a `AA` a `in ` N
`therefore` No such element is prosent in N.
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