prime factorization theorem

Question 1

prime factorization theorem - 1 “Every..”

Answer 1

Every natural number greater than 1 is either a prime number or it can be expressed as a product of prime numbers.

Question 2

prime factorization theorem - 2 “let n…”

Answer 2

Let n be any natural number greater than 1 that is not a prime. Then there must be a factorization of n into two smaller natural numbers both greater than 1 say n=a*b

Question 3

prime factorization theorem - 3 “we now..”

Answer 3

We now look at a and b . If both are primes, we end our proof, since n is equal to a product of two primes. If either a or b is not prime, then it can be factored into two smaller natural numbers, each greater than 1.

Question 4

prime factorization theorem - 4 “if we…”

Answer 4

If we continue in this manner with each of the factors, since the factors are getting smaller, we will fi nd in a fi nite number of steps a factorization of n where all the factors are prime num-bers. So, n is a product of primes, and our proof is complete.