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Question
Check the commutativity and associativity of the following binary operation '*' on N defined by a * b = gcd(a, b) for all a, b ∈ N ?
Solution
Commutativity:
\[\text{Let a}, b \in N . \text{Then}, \]
\[a * b = \gcd\left( a, b \right)\]
\[ = \gcd\left( b, a \right)\]
\[ = b * a \]
\[\text{Therefore},\]
\[a * b = b * a, \forall a, b \in N\]
Thus, * is commutative on N.
Associativity :
\[\text{Let }a, b, c \in N . \text{Then}, \]
\[a * \left( b * c \right) = a * \left[ \gcd\left( a, b \right) \right]\]
\[ = \gcd\left( a, b, c \right)\]
\[\left( a * b \right) * c = \left[ \gcd\left( a, b \right) \right] * c\]
\[ = \gcd\left( a, b, c \right)\]
\[\text{Therefore},\]
\[a * \left( b * c \right) = \left( a * b \right) * c, \forall a, b, c \in N\]
Thus, * is associative on N.
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