Problem 4.51. Let p and q be distinct primes. Find the number of generators of Zpa
8) (Problem 17 (a) on page 49) Let p and q be two distinct primes. Show that for any integer a, pq|(a p+q − a p+1 − a q+1 + a 2 ). Hint: Find the least residue of a p+q − a p+1 − a q+1 + a 2 modulo p, and then find the least residue of a p+q − a p+1 − a q+1 + a 2 modulo q. After that, use the following result: Suppose x,...
7. Let p and q be distinct odd primes. Let a є Z with god(a, M) = 1. Prove that if there exists b E ZM such that b2 a] in Zp, then there are exactly four distinct [r] E Zp such that Zp
6. [Marks 3] Suppose p and q are distinct primes. Find the general solution to the set of equations: x= -1 mod p x = -1 mod q. Show all the steps/details.
(i) State Sylow's theorems. (ii) Suppose G is a group with IGI pr where p, q and r are distinct primes. Let np, nq and nr, denote, respectively, the number of Sylow p, q- and r-subgroups of G. Show that Hence prove that G is not a simple group. (iii) Prove that a group of order 980 cannot be a simple group.
6. Let n be any positive integer which n = pq for distinct odd primes p. q for each i, jE{p, q} Let a be an integer with gcd(n, a) 1 which a 1 (modj) Determine r such that a(n) (mod n) and prove your answer.
1. For n-pg, where p and q are distinct odd primes, define (p-1)(q-1) λ(n) gcd(-1-1.411) Suppose that we modify the RSA cryptosystem by requiring that ed 1 mod X(n). a. Prove that encryption and decryption are still inverse operations in this modified cryptosystem. RSA cryptosystem.
Hello, Can someone please help me proof the following theorem from number theory? thank you! please be legible. 1 11.3.2 LAW OF QUADRATIC RECIPROCITY Restatement) Let p and q be odd primes with p q. Then 1 11.3.2 LAW OF QUADRATIC RECIPROCITY Restatement) Let p and q be odd primes with p q. Then
N=pq with p,q distinct odd primes. Give an expression for the order of (Z/NZ)x in terms of p and q. Then, give an expression for the maximum order of a single element in (Z/NZ)x in terms of p and q.Why does that imply that there does not exist a primitive root modulo N?
Problem 6. Find all primes p such that uoblom7 Tindall
3) Let a -pip?.. .Pk and blp22.. .P%* where pi are distinct positive primes and ri, si are non-negative integers. Then show that where for each i, ni minfri, si) (Recall that min{c, d\ denotes the minimum of c and d.) 3) Let a -pip?.. .Pk and blp22.. .P%* where pi are distinct positive primes and ri, si are non-negative integers. Then show that where for each i, ni minfri, si) (Recall that min{c, d\ denotes the minimum of c...