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

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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| divides 4*3 = 12 6 does not divide either 4 or 3. How can this be? 6 is not prime!

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 = p 1 p 2 …p r and n = q 1 q 2 …q s, where the p's and q's are primes, then r = s and, after renumbering the q's, we have p i = q i for all i.

Sketch of Proof Suppose wlog, r ≤ s. Since p 1 is prime, and p 1 |q 1 q 2 …q s, then by Euclid's lemma, p 1 |q i for some i. Since q i is prime, p 1 = q i. Renumber so that p 1 = q 1. Repeat: p 2 = q 2 … p r =q r. There can be no q's left over, so s = r!