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Number Theory

Number Theory. Lecture 1 Divisibility and Modular Arithmetic (Congruences). Basic Definitions and Notations (1). N = {1,2,3,…} denotes the set of natural numbers Z = {…,-3,-2,-1,0,1,2,3,…} denotes the set of integers. Basic Definitions and Notations (2). Divisibility (1)

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Number Theory

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  1. Number Theory Lecture 1 Divisibility and Modular Arithmetic (Congruences) Peter Burkhardt

  2. Basic Definitions and Notations (1) • N = {1,2,3,…} denotes the set of natural numbers • Z = {…,-3,-2,-1,0,1,2,3,…} denotes the set of integers Peter Burkhardt

  3. Basic Definitions and Notations (2) Divisibility (1) Let a, b e Z, a not equal to zero. We say a divides b if there exists an integer k such that Peter Burkhardt

  4. Basic Definitions and Notations (3) Divisibility (2) In this case we write a|b Sometimes we say that: • b is divisible by a, or • a is a factor of b, or • b is a multiple of a Peter Burkhardt

  5. Basic Definitions and Notations (4) Prime and Composite Numbers A natural number p > 1 is called a prime number, or, simply, prime, if it is divisible only by itself and by 1. P = {2,3,5,7,…} denotes the set of prime numbers. Otherwise the number is called composite. Peter Burkhardt

  6. Properties of Divisibility • a|b ” a|bc for each integer c • a|b and b|c ” a|c • a|b and a|c ” a|(bx + cy) for any x, y e Z • a|b and a, b not equal to zero ” |a| £ |b| Peter Burkhardt

  7. Division with Remainder Let m ,a e Z, m > 1. Then, there exist uniquely determined numbers q and r such that a = qm + r with 0 £ r < m Obviously, m|a if and only if r = 0. Peter Burkhardt

  8. Congruences Let a, b e Z, m e N. We say a is congruent to b modulo m if m|(a-b) and we write Peter Burkhardt

  9. Congruence and Division with Remainder Dividing a and b by m yields the same remainder. Peter Burkhardt

  10. Basic Properties of Congruences That is, congruence is an equivalence relation. Peter Burkhardt

  11. Modular Arithmetic ” Demonstration Peter Burkhardt

  12. Little Fermat’s Theorem Let a e Z, and p prime. If p does not divide a, then For all a e Z we have Peter Burkhardt

  13. Have you understood? How can you write the following statements using congruences? (a, b, r e Z, m e N) • m|a • r is the remainder of a divided by m Using congruences, give a sufficient condition for m|a if and only if m|b Peter Burkhardt

  14. Practice (Handouts) ACTIVITY Peter Burkhardt

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