HOMEWORK 7 SOLUTIONS
Question 1. Let z = x + yi be a complex number (x, y R). Dene the complex
conjugate to be z = x yi, and the absolute value to be |z | = z z = x2 + y 2 .
(1) Prove that
Notation: x is the real part of
HOMEWORK 6 SOLUTIONS
Question 1. Suppose there is a primitive root g modulo n. We demonstrated in
lecture that there are in fact (n) primitive roots. Suppose h is another primitive
root.1 Prove the change of base formula for the discrete logarit
HOMEWORK 4 SOLUTIONS
Question 1. Recall Wilsons theorem states that (p 1)! 1 mod p for
p a prime. Prove that if n > 4 is composite, then (n 1)! 0 mod n. (If
n = 4, then in fact (n 1)! = 6 2 mod 4, the only exception to this 1 or
0 rule I forgot
HOMEWORK 3 SOLUTIONS
Question 1. Suppose that p is a prime and ap + bp = cp for some a, b, c Z.
Show that p|a + b c.
Proof. Fermats Little Theorem tells us that
ap a mod p,
bp b mod p,
cp c mod p
for any prime p. (Note that this is true for all
HOMEWORK 2 SOLUTIONS
(1) Prove that gcd(6n + 5, 7n + 6) = 1 for all n Z.
Solution. Observe that
6(7n + 6) 7(6n + 5) = 1
This is now in the form of Bezouts identity, and allows us to conclude
that gcd(6n + 5, 7n + 6)|1 because any d t
HOMEWORK 1 SOLUTIONS
Question 1 (Stein 1.1). Compute gcd(455, 1235) by hand.
Question 2 (Stein 1.2). Use the Sieve of Eratosthenes to make a list of all
primes up to 100.
Question 3 (Stein 1.3). Prove that there are innitely many primes of the