Fundamental Theorem of Arithmetic

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Presentation transcript:

Fundamental Theorem of Arithmetic

Euclid's Lemma If p is a prime that divides ab, then p divides a or p divides b.

Examples 17 is a prime divisor of 6,052*9,872 Check that 17 | 59,745,344 By Euclid's Lemma 17|6052 or 17|9872. 6 divides 4*3 = 12 6 does not divide either 4 or 3. How can this be? 6 is not prime! 6052 = 17*365 9872 = 17*575+7

Proof of Euclid's Lemma Suppose p does not divide a. Since p is prime, p and a must be relatively prime So there must be integers s,t such that 1 = ps + at. But then b = psb + abt Since p divides both terms on the right, p | b.

Fundamental Theorem of Arithmetic Every integer greater than 1 is a prime or a product of primes. This product is unique, except for the order in which the factors appear. That is, if n = p1p2…pr and n = q1q2…qs, where the p's and q's are primes, then r = s and, after renumbering the q's, we have pi = qi for all i.

Sketch of Proof Suppose wlog, r ≤ s. Since p1 is prime, and p1|q1q2…qs, then by Euclid's lemma, p1|qi for some i. Since qi is prime, p1 = qi. Renumber so that p1= q1. Repeat: p2 = q2 … pr=qr. There can be no q's left over, so s = r!