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32h - Discrete Mathematics Fundamental Theorem of...

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Introduction Fundamental Theorem of Arithmetic Discrete Mathematics Andrei Bulatov Discrete Mathematics – Fundamental Theorem of Arithmetic 32-2 Previous Lecture Common divisors The greatest common divisor Euclidean algorithm Discrete Mathematics – Fundamental Theorem of Arithmetic 32-3 More Primes Prime numbers have some very special properties with respect to division Properties of primes . (1) If a,b are integers and p is prime such that p | ab then p | a or p | b. (2) Let be an integer for 1 i n, and p is prime and then for some 1 i n i a n 2 1 a a a | p K i a | p Discrete Mathematics – Fundamental Theorem of Arithmetic 32-4 The Fundamental Theorem of Arithmetic Theorem . Every integer n > 1 can be represented as a product of primes uniquely, up to the order of the primes. Proof . Existence By contradiction. Suppose that there is an n > 1 that cannot be represented as a product of primes, and let m be the smallest such number. m is not prime, therefore m = st for some s and t But then s and t can be written as products of primes, because s < m and t < m. Therefore m is a product of primes Discrete Mathematics – Fundamental Theorem of Arithmetic 32-5 Example Find the prime factorization of 980,220 Discrete Mathematics – Fundamental Theorem of Arithmetic 32-6 Least Common Multiple
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