Math 453 – Number Theory
Exam 1 October 12, 2005
Make sure to show all of your work on # 1, 2 and 5. Say what theorems you
use to derive your answers.
1. (24 points)
(a) Find [45,54].
(b) Determine how many integers between 1 and 1500 have an inverse modulo 1500.
(c) Find (930012023786827,1000000000000). Do not use the Euclidean algorithm.
2. (20 points)
(a) For what values of cdoes the equation 18x−87y=chave solutions in integers x, y?
(b) Find all solutions of the congruence x4≡4x2(mod p), where pis a prime, p≥3.
3. (13 points) Prove that if 3|aand 4|b, then 12|(8a−9b).
4. (18 points) Prove that if pis a prime and p≥5, then p2≡1 (mod 12). Hint: determine
what remainders are possible when pis divided by 12.
5. (25 points)
(a) Find the remainder when 580 is divided by 7, and the remainder when 580 is divided
by 41.
(b) Use part (a) to find the remainder when 580 is divided by 287 (note that 287 = 7·41).