 Hi and welcome to the session. I am Shashi and I am going to help you to solve the following question. Question is, let star be the binary operation on n defined by a star b is equal to hcf of a and b. Is star commutative? Is star associative? Thus there exist identity for this binary operation on n where n is the set of all natural numbers. First of all let us understand the key idea to solve the given question. A binary operation star on set a is commutative if a star b is equal to v star a for every a b belonging to a. Now a binary operation star from a cross a to a is said to be associative if a star b star c is equal to a star bracket b star c for every a b c belonging to a. Now given a binary operation star from a cross a to a and element e belonging to a if it exists is called the identity for the operation star if a star e is equal to a is equal to e star a for every a belonging to set a. So we have to prove the three conditions that is a star b is equal to b star a and a star b star c is equal to a star bracket b star c and a star e is equal to a is equal to e star a for solving the given question. Let us now start with the solution. First of all let us find if star is commutative. We know for every a b belonging to set of natural numbers a star b is equal to h c f a b similarly v star a is equal to h c f b a. Now we know h c f of a and b is equal to h c f of b and a right. Therefore we can write a star b is equal to b star a. So we can see in key idea a binary operation star on set a is commutative if a star b is equal to b star a. So the given function star in the question is commutative. So we can write star is commutative on n where n is the set of all natural numbers. This completes the first part of the question. Let us now start with the second part of the question that is a star commutative. Now we know for every a b c belonging to n we have a star b star c is equal to h c f a b star c right which is further equal to h c f a b c. Let us now find out a star bracket b star c is equal to a star h c f b c right which is further equal to h c f a b c. So we can see a star b star c is equal to a star bracket b star c since both are equal to h c f of a b and c. Now we can see from key idea that binary operation is associative if a star b star c is equal to a star bracket b star c. So we can say star is associative on n. Let us start with the third part of the question now that is we have to find the identity element. So let us assume e is the identity element a star e is equal to a is equal to e star a right. As we can see from key idea that a given binary operation star from a cross a to a an element e belonging to a if it exists is called identity for the operation star if a star e is equal to a is equal to e star a right of a e is equal to a right h c f of a and a is equal to a is possible only when a is equal to a. Let us now take other element b belonging to set of natural numbers then h c f of b and a must be equal to b right. But this is possible only when e is equal to b. Now we can see in first case our e is equal to a and in second case our e is equal to b. But we know that identity element is always unique and never changes so there does not exist any identity element in the given question. So we can write since there exists no element such that h c f a e is equal to a therefore there does not exist any identity element on n. So our final answer is star is both commutative and associative and star does not have any identity element in n. This completes the session hope you understood the session take care and goodbye.