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?