1. a) i) Find $\operatorname{gcd}(253,189)$ and then find integers $x$ and $y$ such that the equation $253 x+189 y=\operatorname{gcd} (253,189 $ is satisfied. ii) Use the division algorithm to find the quotient $q$ and the remainder $r$ when $-369$ is divided by 8 and hence express $-369$ in the form $-369=8 q+r$. b) Let $\sigma=(1243) (24567) $ be a permutation on the set $S_{7}=\ {1,2,3,4,5,6,7\}$. i) Find the order of $\sigma$. ii) Determine whether $\sigma$ is even or odd. c) Consider the Cayley table of the symmetries of an equilateral triangle $A D C$. \begin{tabular}{ccccccc} & $R_{0}$ & $R_{120}$ & $R_{240}$ & $F_{A}$ & $F_{B}$ & $F_{C}$ \hline$R_{0}$ & $R_{0}$ & $R_{120}$ & $R_{240}$ & $F_{A}$ & $F_{B} $ & $F_{C}$ $R_{120}$ & $R_{120}$ & $R_{240}$ & $R_{0}$ & $F_{C}$ & $F_{A} $ & $F_{B}$ $R_{240) $ & $R_{240) $ & $R_{0}$ & $R_{120) $ & $F_{B}$ & $F_{C} $ & $F_{A}$ $F_{A} $ & $F_{A} $ & $F_{C}$ & $F_{B} $ & $R_{0}$ & $R_{120) $ & $R_{240}$ $F_{B} $ & $F_{B} $ & $F_{A}$ & $F_{C}$ & $R_{240}$ & $R_{0}$ & $R_{120} $ $F_{C}$ & $F_{C}$ & $F_{B}$ & $F_{A} $ & $R_{120}$ & $R_{240}$ & $R_{0}$ \end{tabular) Find the following composites i) $F_{B} F_{C} F_{A}$, ii) $F_{B} F_{C} R_{240}^{2}$, CS.VS. 1393 | 1