100%(1)1 out of 1 people found this document helpful
This preview shows page 1 - 5 out of 18 pages.
MATH150: Chapter 4 Page 1 MATH 150 Chapter 4: Number Theory and Cryptography4.1 Divisibility and Modular Arithmetic 4.2 Integer Representations and Algorithms 4.3 Primes and Greatest Common Divisors 4.4 Solving Congruences
MATH150: Chapter 4 Page 2 4.1 Divisibility and Modular ArithmeticDivisionDEFINITION 1such that of a. b. b. EXAMPLE Determine whether 3|7 and whether 3|12. EXAMPLE Let n and d be positive integers. How many positive integers not exceeding n are divisible by d? ≤ n/d. d. (b + c); c; a | c. | c. c = at. a(s + t). c.
MATH150: Chapter 4 Page 3 COROLLARY 1a | c, dq + r. DEFINITION 2 In the equality given in the division algorithm, d is called the divisora is called the dividendq is called the quotientand r is called the remainderThis notation is used to express the quotient and remainder: q = a div d, r = a EXAMPLEWhat are the quotient and remainder when 101 is divided by 11? , , , . mod d. . 11. EXAMPLEWhat are the quotient and remainder when −11 is divided by 3? . 3, 3. 2,3. Modular ArithmeticDEFINITION 3 if m b (plural m).
Page 4 THEOREMLet a and b be integers, and let m be a positive integer. Then