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(D) Given,in a group $(Z, *)$,the operation is defined as $a * b = a + b - n$ for all $a, b \in Z$. First,we find the identity element $e$ such that $

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$3n$

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(D) Given,in a group $(Z, *)$,the operation is defined as $a * b = a + b - n$ for all $a, b \in Z$. First,we find the identity element $e$ such that $a * e = a$. $a + e - n = a \implies e = n$. Now,to find the inverse of $(-n)$,let the inverse be $x$ such that $(-n) * x = e$. Since $e = n$,we have $(-n) * x = n$. Using the definition of the operation: $(-n) + x - n = n$. $x - 2n = n$. $x = 3n$. Thus,the inverse of $(-n)$ is $3n$.

In the set of integers $(Z, )$,if $a b = a + b - n, \forall a, b \in Z$,where $n$ is a fixed integer,then the inverse of $(-n)$ is:

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In the set of integers $(Z, *)$,if $a * b = a + b - n, \forall a, b \in Z$,where $n$ is a fixed integer,then the inverse of $(-n)$ is: