There's a question in my textbook:
Let * be the binary operation on N defined by a*b = HCF of a and b. Does there exist identity for this binary operation on N?
According to me the answer should be yes, since a*(any multiple of a)=a. However, the answer is given as no in the answers.
What is the answer?
The identity must work for every element: you need some number $e$ such that $$ e*a=a $$ for all $a$.
In particular, $e$ should be a multiple of $a$, for every $a$.
Such a number does exist and is $0$, if you allow it in the natural numbers and also define $\gcd(0,0)=0$. Some people don't list $0$ among the natural numbers, some others don't define $\gcd(0,0)$. So the answer depends on the conventions your textbook uses. Personally I see no reason for avoiding $0$, but others don't agree.