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Let $\ast$ be the binary operation on $N$ given by $a \ast b =$ LCM of $a$ and $b$. $(iii)$ Is $ \ast$ associative?

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  • 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 operation $\ast$ on $A$ is associative if $a\ast ( b \ast c) = (a \ast b) \ast c\; \forall \; a, b, c \in A$
Given that the binary operation on N given by $a \ast b = $ LCM $(a,b)$:
$\Rightarrow (a \ast b) \ast c = ($LCM $(a,b)) \ast c$ = LCM ($a,b,c)$
$\Rightarrow a \ast( b \ast c )= a \ast ($LCM $(b,c) =$ LCM $(a,b,c)$
Since $ (a \ast b) \ast c = a \ast( b \ast c), \; \ast$ is associative.
answered Mar 19, 2013 by balaji.thirumalai

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