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!

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!