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3 Prime Factorization Skills

More Notes

Recognition of prime factors and the prime decomposition of whole numbers speeds calculations of LCM, GCD and LCD in exact arithmetic with whole numbers and fractions. They also lead to cosmetic simplification of square roots. The recognition of common factors in numerators and denominators of fractions alone or in products helps with the reduction of fractions and via cancellation leads to efficient methods for multiplying fractions.

Description of Folder Lessons

1. [video] how Products are bigger than factors. This observation simplifies the identification of primes and composite numbers.

2. Prime or Composite less than 16.. Instead of saying a whole number is prime when and only when it is not a product of whole numbers larger than one, we say a whole number is prime when and only when it is not the product of two or more smaller whole numbers larger than one. The addition of the word smaller, redundant and technically not required due to lesson 1, none the less makes prime number identification easier to learn and teach. This webpage employs the definition of primes and the 12 times table to recognize that the whole numbers 2, 3, 5, 7, 11 and 13 as primes.

3. [video] Primes and Composites from 9 times table. Saying a whole number is prime when and only when it is not the product of two or more smaller whole numbers larger than one allow small primes in the 9 times table to be quickly identified. Here is a short form video version or variant of the previous lesson.

4. [video] Prime Factorization Introduction. This video lesson introduces and illustrates prime factorization for a few whole numbers. In the last example, the equivalence between tree notation and the use of equal signs in the development of prime factorization is emphasized.

5. Prime Factorization and a Square Rule. This lesson provides a more detailed discussion of prime factorization and introduces a well-known, but I suspect hitherto nameless rules, for identifying primes and obtaining prime factorization. The name square rule is coined. This rule or method provides students with a quick manual method for obtaining the prime factorization of whole numbers. This quick method can be employed with the aid of the divisibility rules for the small primes 2, 3, 5 and 11, when or if mastered, or with the aid of calculators that display sufficiently many digits after the decimal point. See the calculator usage notes and cautions below.

6. Sieve-of-Eratosthenes upto 100. Discussion of this Sieve (or filter) shows how striking out of multiples of 2, 3, 5 and 7 is sufficient by the square rule to identify all primes in a list or table of whole numbers 1 to 100.

7. Calculator Usage Notes and Cautions. The square rule for identifying primes and obtaining prime factorizations, two side of the same coin, can be employed with the aid of calculators that display sufficiently many digits after the decimal point. Here are notes on how with some cautions.

8. [video] Prime Factorization upto 19. Given the identification of all primes less than 100, this video is redundant. However, it is retained to illustrate the use of the square rule. An alternative approach would be to use the 18 times table alone or a subtable of a larger times table.

9. [video] Prime Factorization upto 19 squared. The square of 19 is 361. provides examples of prime factorizaton with the square rule for a few numbers less 361.

10. [video] Prime Factorization upto 23 squared. The square of the prime 23 is 5629. provides examples of prime factorizaton with the square rule for a few numbers less 529.

11. Efficient Square Rule Use. The last part of the lesson Prime Factorization and a Square Rule provides a few hints or directions for the efficient use of the square rule. This lesson on efficient use provides examples.

12. LCD GCD and LCM calculations using Primes. Finding least ommon ddominators, least ommon multiples and greatest common divisors are of sets of whole numbers, two or more, represent number theory practices of service in simplifying, adding, comparing, subtracting, multiplying and dividing fractions, and of service in cosmetic conventions for the exact representation of square, cube and higher roots of whole numbers. This webpage gives some examples of LCD GCD and LCM calculations using Primes, or more precisely prime factorizations.

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    Lessons 13 to 18 present and illustrate methods to count and find all whole number factors of a given whole number using tables, products and trees. In the study of quadratics, factoring by inspection requires the identification of all integer factors of a given integer. The methods here or variant of them turn that identification part of this factoring a quadratic by inspection into a systematic art.

13. [video] Factors of 24 using primes

14. [video] Factors of 24 Take II

15. [video] Factors of 20 using Prime Factorization

16. [video] Factors of 980 using primes

17. Identify and Count Factors using Primes

18. [video] Count Factors given Prime Factorization

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19. [video] Prime Factorization Unique. This video lesson introduces the notion that the prime factorization of a whole number will be independent of the route that gives the factorization. Knowing about the uniqueness and how it implies the latter is nuance that may included for completeness, or not.

20. Uniqueness of Prime Factorization. This lesson describes the uniqueness in general - offers reason for it.

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science