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30h - Discrete Mathematics Primes 30-2 Integers"God made...

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Introduction Primes Discrete Mathematics Andrei Bulatov Discrete Mathematics - Primes 30-2 Integers God made the integers; all else is the work of man Leopold Kroenecker Discrete Mathematics - Primes 30-3 Most of useful properties of integers are related to division If a and b are integers with a 0, we say that a divides b if there is an integer c such that b = ac. When a divides b we say that a is a divisor ( factor ) of b, and that b is a multiple of a. The notation a | b denotes that a divides b. We write a | b when a does not divide b Example. Let n and d be positive integers. How many positive integers not exceeding n are divisible by d? The numbers in question have the form dk, where k is a positive integer and 0 < dk n. Therefore, 0 < k n/d. Thus the answer is n/d Division Discrete Mathematics - Primes 30-4 Properties of Divisibility Let a, b, and c be integers. Then (i) if a | b and a | c, then a | (b + c); (ii) if a | b, then a | bc for all integers c; (iii) if a | b and b | c, then a | c. Proof. (i) Suppose a | b and a | c. This means that there are k and m such that b = ak and c = am. Then b + c = ak + am = a(k + m), and a divides b + c. Discrete Mathematics - Primes 30-5 Properties of Divisibility (cntd) If a, b, and c are integers such that a | b and a | c, then a | mb + nc whenever m and n are integers. Proof. By part (ii) it follows that a | mb and a | nc. By part (i) it follows that a | mb + nc. If a | b and b | a, then a = ± b. Proof. Suppose that a | b and b | a. Then b = ak and a = bm for some integers k and m. Therefore a = bm = akm, which is possible only if k,m = ± 1. Discrete Mathematics - Primes 30-6 The Division Algorithm Theorem (The division algorithm) Let a be an integer and d a positive integer. Then there are unique integers q and r, with 0 r < d, such that a = dq + r d is called the divisor , a is called the
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