chapter4 - MATH 150 Chapter 4 Number Theory and Cryptography 4.1 Divisibility and Modular Arithmetic 4.2 Integer Representations and Algorithms 4.3

# chapter4 - MATH 150 Chapter 4 Number Theory and...

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MATH150: Chapter 4 Page 1 MATH 150 Chapter 4: Number Theory and Cryptography 4.1 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 Arithmetic Division DEFINITION 1 such 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 1 a | 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 Arithmetic DEFINITION 3 if m b (plural m ).
Page 4 THEOREM Let a and b be integers, and let m be a positive integer. Then