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

$\begin{array}{1 1} \text{Only 1} \\ \text{1 and 0} \\ \text{Any element in N} \\ \text{Only a and b} \end{array}$

Toolbox:
• The lowest common multiple of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b.
• An element $a \in N$ is invertible if then exists an element b in N such that $a \ast b=b\ast a = e$ where e is identify element $[e=1 \in N]$
Given that the binary operation on N given by $a \ast b =$ LCM $(a,b)$:
$e=1 \rightarrow \;$ LCM $(a,b)$ = LCM $(b,a) = 1$.
This is possible only if $a = b = 1 \rightarrow 1$ is the only invertible element in $N$.
edited May 4, 2016 by meena.p