Let $^*$ be the binary operation on $N$ given by $a \,^*\, b = \text{L.C.M. of } a \text{ and } b$. Is $^*$ associative?
(A) For $a, b, c \in N,$
$(a \,^*\, b) \,^*\, c = (\text{L.C.M. of } a \text{ and } b) \,^*\, c = \text{L.C.M. of } a, b, \text{ and } c$
$a \,^*\, (b \,^*\, c) = a \,^*\, (\text{L.C.M. of } b \text{ and } c) = \text{L.C.M. of } a, b, \text{ and } c$
$\therefore (a \,^*\, b) \,^*\, c = a \,^*\, (b \,^*\, c)$
Thus,the operation $^*$ is associative.
$(a \,^*\, b) \,^*\, c = (\text{L.C.M. of } a \text{ and } b) \,^*\, c = \text{L.C.M. of } a, b, \text{ and } c$
$a \,^*\, (b \,^*\, c) = a \,^*\, (\text{L.C.M. of } b \text{ and } c) = \text{L.C.M. of } a, b, \text{ and } c$
$\therefore (a \,^*\, b) \,^*\, c = a \,^*\, (b \,^*\, c)$
Thus,the operation $^*$ is associative.
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