Question

In the set $N$ of natural numbers, define the binary operation $*$ by $m * n = GCD ( m , n ), m , n \in N$. Then, which of the following is true?
I. $*$ is not a binary operation
II. $^{*}$ 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 $^{*}$ is a binary operation, as $*$ 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 $*$
i.e., $a^{*} e = a = e ^{*} a \forall a \in N$
$\Rightarrow GCD ( a , e )= a = GCD ( e , a )$
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 $^{*}$ does not exist.
Hence inverse of elements of $N$ does not exist.