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# Let ∗ be the binary operation on N defined by a ∗ b = H.C.F. of a and b. Is ∗ commutative? Is ∗ associative? Does there exist identity for this binary operation on N?

Q. 8 Let ∗ be the binary operation on N defined by a ∗ b = H.C.F. of a and b.
Is ∗ commutative? Is ∗ associative? Does there exist identity for this binary
operation on N?

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a ∗ b = H.C.F. of a and b for all $a,b \in A$

H.C.F. of a and b = H.C.F of b and a for all $a,b \in A$

$\therefore \, \, \, \, a*b=b*a$

Hence, operation  ∗  is commutative.

For  $a,b,c \in N$ ,

$(a*b)*c = (H.C.F \, of\, a\, and\, b)*c= H.C.F\, of \, a,b,c.$

$a*(b*c )= a*(H.C.F \, of\, b\, and\, c)= H.C.F\, of \, a,b,c.$

$\therefore$                $(a*b)*c=a*(b*c)$

Hence,  ∗ is associative.

An element $c \in N$ will be identity for operation *  if $a*c=a= c*a$  for $a \in N$.

Hence, the operation * does not have any identity in N.

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