Question

Let $\ast$ be the binary operation on N given by $a\ast b=LCM$ of a and b.
(i) Which elements of N are invertible for the operation $\ast$?

Answer 1

The binary operation $\ast$ on N is defined as $a\ast b=LCM$ of a and b.
An element a in N is invertible with respect to the operation $\ast$ if there exists an element b in N such that $a\ast b=e=b\ast a$.
Here, e=1, This means that
LCM of a and b =1 =LCM of b and a
This case is possible only when a and b are equal to 1.
Thus, 1 is the only invertible element of N with respect to the operation $\ast .$