Using the basic form of the Euclidean algorithm, compute the greatest common divisor of $(1)16308$ and $11382(10$ points$)$; $(2)2395$ and $455(10$ points$).$

With the Extended Euclidean algorithm, we finally have an efficient algorithm for finding the modular inverse. Figure out whether there are the inverses of the following $x$ modulo $m.$ If yes, please use EEA to calculate it$.$ If not, please explain why.

$($a$)x=7,$ Modular Exponentiation:

Please compute $7^{13}$ mod $15$ manually and show all the calculation procedures.

$m=17(5$ pomits$)$

$($b$)x=13,m=85(10$ points$)$

$($c$)x=4,m=12(5$ points$)$