# 4. i) Suppose n = pq where p and q are distinct odd primes. Let r be an integer such that r = p^-1 (

4. i) Suppose n = pq where p and q are distinct odd primes. Let r be an integer such that r = p^-1 (mod q), and put s = 1 – 2rp. Let a be an integer such that (a, n) = 1. Show that the solutions modulo n to the congruence X^2 = a^2 (mod n) are precisely x = + or – a (mod n) and z + or – as (mod n) (you may use results from the previous problem set). ii) Find all solutions to x^2 = 49 (mod 53 . 83).

Show transcribed image text 4. i) Suppose n = pq where p and q are distinct odd primes. Let r be an integer such that r = p^-1 (mod q), and put s = 1 – 2rp. Let a be an integer such that (a, n) = 1. Show that the solutions modulo n to the congruence X^2 = a^2 (mod n) are precisely x = + or – a (mod n) and z + or – as (mod n) (you may use results from the previous problem set). ii) Find all solutions to x^2 = 49 (mod 53 . 83).