Question

Let $A=\{1,2,3,4,5,6\}$ and ${ }^*$ be an operation $A$ defined by $a{ }^* b=r$, where $r$ is the least non-negative remainder when the product $a b$ is divided by $k$. Operation * is binary operation if $k=$

Answer 1

$1(\mathrm{~A})=1=0(7)+1 \Rightarrow 1 * 1=1 \in A$

$1(B)=2=0(7)+2 \Rightarrow 1^* 2=2 \in A$

$1(C)=3=0(7)+3 \Rightarrow 1^* 3=3 \in A$

$5(5)=25=3(7)+4 \Rightarrow 5^* 5=4 \in A$

$5(6)=30=4(7)+2 \Rightarrow 5^* 6=2 \in A$

Also, by the definition of ${ }^{\prime * \prime}$,

We have $\quad a^* b=b^* a \forall a, b \in A$

$\therefore \quad a^* b \in a, b \in A$

$\therefore *$ is a binary operation on $A$.