The Fundamental Theorem of Arithmetic
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• 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, and
2. 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 = 2 2 2 2 2 5 5 7 = 25 52 7
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.