Quadratic congruences - 8. Find all solutions to the quadratic congruences, if they exist.
(a) x2 + x + 1 ≡ 0 (mod 7).
(b) x2 ≡ 55 (mod 179)
Congruences and Primes - Prove that is p is prime, we have:
n choose m is congruent to [floor(n/p) choose floor(m/p)]*[(n mod p) choose (m mod p)] (mod p)
Hint: show that (1+x)^(pq+r) is congruent to (1+x)^r * (1+x^p)^q ...
Systems of Congruences : Simultaneous Solutions - 1. Prove that gcd (a, lcm[b, c]) = lcm[gcd(a,b), gcd(a,c)].
2. Find the simultaneous solutions of the following congruences:
2x ≡ 1(mod 5)
3x ≡ 9 (mod 6)
4x ≡ 1 (mod 7)
...
Discrete Structures : Congruences - 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 t ...
Quadratic Congruences - (See attached file for full problem description)
Determine which of the following quadratic congruences has solutions:
a.
b.
c.