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Understanding Prime Factorization and Counting Factors: The Fundamental Theorem of Arithmetic

This resource explores the Fundamental Theorem of Arithmetic, which states that every natural number greater than 1 can be uniquely expressed as a product of prime numbers. It discusses prime numbers, composite numbers, and offers insights on counting the factors of a number. We will examine methods for determining primality and look at interesting patterns in remainders from division by small primes. Additionally, we highlight useful calculator programs for finding prime factorizations and applying the Euclidean algorithm.

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Understanding Prime Factorization and Counting Factors: The Fundamental Theorem of Arithmetic

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  1. Part II – Theory and Computations

  2. Major Ideas....

  3. 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).

  4. 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

  5. The numbers 2, 3, 5 and 7 determine primes < 100 5

  6. The numbers 2, 3, 5 and 7 determine primes < 100 5 5 The primes above determine all primes < 10,000

  7. Counting Factors... How can we count the factors of a number? For example: How many factors does 180 have?

  8. Counting Factors...

  9. Proofs....(The ones that can be explained to interested students.)Infinitely many primes, determining whether a number is prime, and divisibility by 3.

  10. Another Look...(with a possibility for generalization)

  11. Can you see what is happening in the following slide?In particular, what are the blue numbers?

  12. The blue numbers are the remainders from dividing 1, 10, 100, 1000, 10000, etc. by 3

  13. Can you see what is happening in the following slide?In particular, what are the blue numbers?

  14. Notice the repetition!!

  15. The blue numbers are the remainders from dividing 1, 10, 100, 1000, 10000, etc. by 7

  16. Can you state a give a conjecture concerning divisibility by 7?

  17. 1, 3, 2, 6, 4, 5, 1, 3, 2, 6, 4, 5, . . .

  18. Can you find the pattern for divisibility by 11?

  19. Can you find the pattern for divisibility by 11? Answer: 1,10,1,10,1,10,...

  20. Computations...

  21. Most calculators and computers use the Euclidean algorithm.

  22. TI-83 Programs... iPart, gcd and lcm – TI-83 commandsPCHECK – (program) determines whether a number is prime.PFACT – (program) gives the prime factorization of a number.

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