Part II – Theory and Computations
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Transcript of Part II – Theory and Computations
Fundamental Theorem of Arithmetic(Prime Factorization Theorem)
Each natural number n can be written as a product of prime numbers in one and only one way (except for the order of the factors).
THEOREM: If n > 2 is a composite number, then n has a prime divisor p such that
CORLLARY: If n > 2 has no prime divisors p such that , then n is a prime number.
APPLICATION: To test whether a number n is a prime you only have to check whether n is divisible by the primes
.np
np
.np
Counting Factors...
How can we count the factors of a number?
For example: How many factors does 180 have?
Counting Factors...
1
1 2
1
1
1
Let with 2. If , ,...,
are distinct prime numbers and
,..., 1, 2,... so that
, then there are
1 1 factors of .
k
k
k
k
k
n n p p p
n p p
n
Proofs....
(The ones that can be explained to interested students.)
Infinitely many primes, determining whether a number is
prime, and divisibility by 3.