# Is $$\ast$$ defined on the set $$\{1,2,3,4,5\}$$ by $$a\ast b=L.C.M$$.of $$a\;and\;b$$ a binary operation?

$\begin{array}{1 1} \text{at is a binary operation} \\ \text{at is not a binary operation } \end{array}$

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.
• A binary operation $\ast$ on a set $A$ is a function $\ast$ from $A \times A$ to $A$. Therefore, if $a,b \in A \Rightarrow a \ast b \in A\; \forall\; a,b, \in A$
Given $\ast$ on set $A: \{1,2,3,4,5\}$ is defined by $a \ast b =$ LCM $(a,b)$:
Consider $a=1, b=2 \rightarrow 1 \ast 2 =$ LCM $(1,2) = 2$ and $2 \in A$.
However, if we consider another pair of elements $a=2, b=3 \rightarrow 2 \ast 3 =$ LCM$(2,3) = 6$ and $6 \not \in A$
Therefore, $\ast$ is not a binary operation.