$($a$)$ Find the solution to the following simultaneous equations using chinese remainder theorem

$x-=2$mod$6,x-=1$mod$5,x-=3$mod$7$

$1$

Figure $1$: Calculation of message digest

$($b$)$ In a public key cryptosystem using RSA, an adversary intercepts the ciphertext $C=5$ sent to a user whose public key is $e=7,$ and $n=35.$ What is the plaintext $P?$

$($c$)$ Test the primality of the number $53$ using Miller$-$Rabin primality test $($take at most three values of $a,$ if required, i$.$e$.,$ choose $a=2,a=3,a=4).$