Solving Systems of Congruences - Find all solutions to the system of congruences:
x ≡ 2 (mod 3)
x ≡ 1 (mod 4)
x ≡ 3 (mod 5)
Solving Linear Congruences - 1.66
Find all solutions x to each of the following congruences:
(I) 3x 2 mod 5
(II) 7x 4 mod 10
(III) 243x + 17 101 mod 725
(IV) 4x + 3 4 mod 5
(V) 6x + 3 4 mod 10
(VI) 6x ...
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)
...