Construct addition and multiplication tables for arithmetic modulo 11. For example, 7 + 8 mod 11 is
4 and 7*8 mod 11 is 1 so the entry in row 7, column 8 would be 4 for the addition table and 1 for the multiplication table.
Use your tables to solve each of the following congruences:
a. 3x+2≡8 (mod11)
b. 3x-5≡2 (m
6. Let g be a primitive root of m. An index of a number a to the base (written ing a) is a number + such that g+≡a(mod m). Given that g is a primitive root modulo m, prove the following...
7. Construct a table of indices of all integers from....
8. Solve the congruence 9x≡11(mod 17) using the table in 7.
The idea of this problem is to investigate solutions to x2≡1(mod pq) where p and q are distinct odd primes.
(a) Show that if p is an odd prime, then there are exactly two solutions modulo p to x2≡1(mod p).
(b) Find all pairs (a,b) Є Zp x Zq such that a2≡1(mod p) and b2≡1(mod q).
(c) Let p=17 an
Quadratic Congruences. See attached file for full problem description.
Let p be an odd prime. Complete the proof of the question "For which odd primes p is LS(2, p) = 1?" by showing that if and that if .
Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields.
(See attached file for full problem description)