THEORY OF NUMBERS
HOMEWORK 1 SOLUTIONS
Computations (1) (a) Compute the divisors of both 18 and 24 (don't forget negative divisors). (b) Find the greatest common divisor of 18 and 24, that is, the largest integer that is a divisor of both 18 and 24. (c) C

THEORY OF NUMBERS
HOMEWORK 5 SOLUTIONS
Computations (1) Compute the following which are related to the Mbius function: o (a) Compute the values of the Mbius function (n) for n = 1, , 20 o (b) In class, we showed that (n) = d|n d and (n) = d|n 1. Use the M

THEORY OF NUMBERS
HOMEWORK 3 SOLUTIONS
Computations (1) Use the Chinese remainder theorem to find a solution to each of the following linear systems: (a) x1 x2 x3 (b) x0 x0 x1 x6 (c) x2 x3 x4 x5 x6 mod mod mod mod mod 11 12 13 17 19 mod mod mod mod 2 3 5

THEORY OF NUMBERS
HOMEWORK 4 SOLUTIONS
Computations (1) (a) Compute the prime factorizations of (i) 289 (ii) 256 (iii) 714 (iv) 5040 (b) Use the prime factorizations computed above to compute (i) gcd(289, 714) (ii) gcd(256, 5040) (iii) gcd(714, 5040) (iv)

THEORY OF NUMBERS
HOMEWORK 2 SOLUTIONS
Computations (1) (a) Compute the following: gcd(-27, -45), gcd(100, 121), and gcd(1001, 289) using the Euclidean algorithm. (b) For each of the pairs in part (a), find the linear combination of these integers equal t

THEORY OF NUMBERS
HOMEWORK 6 SOLUTIONS
Computations (1) Find all solutions of the following congruences: (a) x2 58 mod 77 (b) x2 207 mod 1001. Solution: (a) Note that 77 = 7 11. Since if x0 solves the polynomial x2 58 mod 77, it also solves the polynomial

THEORY OF NUMBERS
HOMEWORK 10 SOLUTIONS
Computations (1) Determine if the Gaussian integer divides the Gaussian integer in the following pairs: (a) = 3 and = 4 + 7i (b) = 5 + 3i and = 30 + 6i (c) = 2 + i and = 15 (d) = 11 + 4i and = 274 Solution: (a) Sinc

THEORY OF NUMBERS
HOMEWORK 9 SOLUTIONS
Computations (1) Find all solutions of each of the following congruences: (a) x3 + 8x2 - x - 1 0 mod 11 (b) x3 + 8x2 - x - 1 0 mod 121 (c) x3 + 8x2 - x - 1 0 mod 1331 Solution: (a) Since 11 is a prime, there is no be

THEORY OF NUMBERS
HOMEWORK 7 SOLUTIONS
Computations (1) For each of the integers 10, 12, 14, 18, 20 either find a primitive root or show that none exists. Solution: (a) 3 is a primitive root mod 10 since (10) = (2)(5) = 4 and 31 3 mod 10, 32 9 mod 10, 33