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Appetizers and Lessons for Mathematics and Reason 
a calculus and preparation for calculus website, etc.

||Définition d'une variable || Algèbre || Arithmetique || Logique ||La raison basée sur les règles et modelés||
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Read  logic chapters 1 to 5  in online volume Three Skills for Algebra  for greater skills & confidence in  work 
and study.

Learn to read notes and textbooks like a lawyer, so that no nuance, no subtlety and no clause escapes your attention.

Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought

1. [Play Video] 5 minutes - A Times Table (10 x 10) and how a number is not prime (composite) if it is in the interior of the table, that is if it is a product of smaller natural numbers. Some where in here is a Definition for Primes. A Natural number is composite if it is not prime.
   
2. [Play Video] 9½ minutes - Digit- Based Rules for recognizing divisibility by the divisors 2, 3, 5, 9, 10 and 11 or  calculating the remainders on division by these divisors. These rules follow from  10 = 0 mod 2 or 5, and 10 = 1 mod 3 or 9, and 10 = -1 mod 11.  Exercise: (1) Use  100 = 2 mod 49 to develop a digit-based rule for division by 49 or 7. (2) Give digit-based rules for division by 2, 3,5, 7, 11 and 13 that apply to the hexadecimal representation of whole numbers.
   
3. Square Root Rule: A number N is prime if it is not divisible by all primes p whose square p² is less than or equal to N.  On the other hand if a number N is not prime, it will be divisible by a prime p with p² less than N+1. With a calculator, the best bet is check where all primes p < sqrt(N) starting with the smallest.  Here if N = Mq where all primes < p are not divisor of the prime N then all primes < p will not be divisors of M. With the aid of a calculators and rules for divisibility by 2,3, 5, and 11, you can quickly get the prime decomposition of a whole number N.
   
4. [Play Video] 10 minutes - Recognizing Primes in the interval to 100 by eliminating all numbers that are multiples of primes < 11 = the first prime with square 11² = 121 > 100. (The Sieve of Erasothenes)
   
   If a first number N is a product of two factors, the square of the  larger factor will be greater than or equal  to the first number, and the square of the smaller will be less than or equal the first number N. So if the first number N can be factored, there will be a divisor, the smallest factor in a product with square < the first number N. That in turn implies there will be a prime <  the smallest factor which divides N and whose square is  <  N. From the study of logic (the contrapositive of an implication rule), if all primes with square < N do not divide N, N cannot be written as a product of factors - natural numbers smaller than N.
   
5. [Play Video] 2½  minutes -  Prime Factorizations (also called decomposition) for numbers in the interval 2 to 15.
   
6. [Play Video] 3     minutes - Prime Factorizations  for numbers in the interval 16 to 30.
   
7. [Play Video] 4½  minutes - Prime Factorizations for numbers in the interval 31 to 49.
   
8. [Play Video] 4     minutes - Prime Factorizations  for numbers in the interval 50 to 66. Note: 51 = 3 x 17 is not prime as stated in video. Oops.
   
9. [Play Video] 5½  minutes - Prime Factorizations for numbers in the interval 67 to 82.
   Note: 76 = 2 x 38 = 2 x 2 x 19. Video shows 17 instead of 19. Oops
   
10. [Play Video] 5½  minutes -  Prime Factorizations  for numbers in the interval 83 to 100.
    Note: 90 = 6 x 15 = 2 x 3 x 3 x 5 = 2 3² 5 Video write 4 x 15 instead of 6 x 15. Oops