$1.$ Consider the following version of the Euclidean algorithm to compute $(,)$: Step $1$: Swap the numbers if necessary to have $<=$; Step $2$: If $=0,$ then return $$; Step $3$: If $!=0,$ then replace $$ by $$ and go to step $(1).($a$)$ Carry out this algorithm to compute $(51,2)($b$)$ Show that the modified Euclidean algorithm always terminates with the right answer $($c$)$ How long does this algorithm take, in the worst case, when applied to two $100-$digit integers?