intsu11h3

intsu11h3 - Introductory Number Theory. Homework 3 Due:...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Introductory Number Theory. Homework 3 Due: Tuesday, June 7, 2011, 4:45PM 1. Let m,n be integers, not both 0. Consider the set A = { mx + ny : x,y Z } . Show that this set coincides with the set of all multiples of the greatest common divisor of m,n . That is, if d = gcd ( m,n ), show A = { kd : k Z } . 2. If a,m N and gcd ( a,m ) = 1, then the order of an integer a modulo m is the smallest positive integer k such that a k 1 (mod m ). Prove: If gcd ( a,m ) = 1 and if k is the order of a modulo m , then k | φ ( m ). Hint: Divide φ ( m ) by k ; what can you say about the remainder? 3. Assume p 3 is prime and that p 6 | a and a 6≡ 1 (mod p ). Prove: p - 2 X k =1 a k = a + a 2 + · + a p - 2 ≡ - 1 (mod p ) . (of course, you do not have to prove the equality; that’s just the definition of the summation expression; you only have to prove the congruence.) 4. Textbook, Exercise 13.3 (p. 89). 5. Textbook, Exercise 14.1 (p. 94).
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 2

intsu11h3 - Introductory Number Theory. Homework 3 Due:...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online