Question

In the set $N$ of natural numbers, define the binary operation $\ast$ by $m\ast n=GCD(m,n),m,n\in N$. Then, which of the following is true?
I. $\ast$ is not a binary operation
II. ${}^{\ast }$ is a binary operation
III. Inverse of each element of $N$ exist
IV. Inverse of each element of $N$ does not exist

• A. I and IV are true
• B. II and III are true
• C. Only I is true
• D. II and IV are true

Answer 1

Answer: (D)

Clearly ${}^{\ast }$ is a binary operation, as $\ast$ carries each pair $(m,n)\in N\times N$ to a unique element $GCD(m,n)$ in $N$.
Now, in order to find the inverse of elements of $N$,
let us first find the identity element if exist.
Let $e\in N$ be the identity element for $\ast$
i.e., $a^{\ast }e=a=e^{\ast }a\forall a\in N$
$\Rightarrow GCD(a,e)=a=GCD(e,a)$
Note that $GCD(a,e)=a$ iff $e=a$ or $e$ is a multiple of $a$.
Thus, identity element is not unique.
$\therefore$ Identity element for ${}^{\ast }$ does not exist.
Hence inverse of elements of $N$ does not exist.