1. By: Katrina Carlsen and Melissa Sparow

2.  For every integer n ≥ 2, n is a prime number or can be written uniquely (ignoring ordering) as a product of prime numbers  Ex: unique factorization of 825  Divide by successively larger primes (2,3,5 etc)

3. Prove the factorization of a composite number n>2 into prime numbers is unique when order is not important. N=p 1 p 2 p 3 …p r and N=q 1 q 2 q 3 …q s p 1 p 2 p 3 …p r = q 1 q 2 q 3 …q s P 1 | p 1 p 2 …p r so P 1 | q 1 q 2 …q s This means P 1 | q i Contradiction. This means factorization is unique!

4.  integers a and b are relatively prime if gcd(a,b) = 1.  ia +jb = 1  i(21) + j(16) = 1 Find i and j   i (27) + j(25) = 1

5.  p|p and p|a  gcd(a,p)= p  gcd(a,p) = 1  p|ab  P must divide either a or b  P doesn’t divide a..  1 = ia + jp  Linear combination of prime #s

6. Given positive integers a and b, gcd(a,b) is the linear combination of a and b that has the smallest positive value.

7.  1 = ia + jp  Multiply by b  b = (ia)b + (jp)b = i(ab) + (jp)b  p|ab… ab written as kp  b = i(kp) + (jp)b  b = (ik +jb)p  p|b PROVEN!

8.  Our goal was to proof the Fundamental Theorem of Arithmetic  In order to do so, we needed to understand the Theorem on Division by Prime Numbers, what it means to be Relatively Prime, Linear Combinations and the Theorem on gcd(a,b).

9. Theorem on Size of Prime Factors: If n is a composite number, then it has a prime factor less than or equal to √n. Given n=1021, find the prime factors of n or determine that it is prime. √1021 = 31.953 So we will test with prime numbers: 2,3,5,7,11, 13,17,19,23,29,31 None divides 1021. That means 1021 is prime.