The Fundamental Theorem of Arithmetic (2/12)

• Definition (which we all already know). A number greater than 1 is called prime if its only divisors are 1 and itself. Otherwise it is called composite.

• Lemma. If a prime number p divides the product a b of two integers, then p divides a or p divides b (or both)

• Note that the condition on p that it be prime is necessary. For example, 6 | (10)(15), but 6 10 and 6 15.

• The proof of this lemma uses the EEA! If p | a were done, so suppose p a. Hence p and a must be relatively (why?). Now apply the EEA to p and a and proceed.

The Theorem• The Fundamental Theorem of Arithmetic. Every number

greater than 1 can be written uniquely as a product of prime numbers.

• Note: This says two things: 1. Every number above 1 can be so written, and2. the representation is unique (up to the order of the factors).

• Note: We normally write the factorization in order of smallest prime to largest prime, and we also gather multiple occurrences of any single prime into one occurrence with an exponent.

• Example: 5600 = 22222557 = 25527

Proof of the Theorem• The proof of existence is by induction. Discuss.• The proof of uniqueness is by the Lemma on the first

slide, which is due to the EEA.• For Friday, read Chapter 7 and do Exercises 7.1 and 7.3.