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YOU are better than YOU think. Show
yourself how:
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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. |
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Logic
chapters 1 to 5 re- appear not in sequence, as is or longer,
in Volume 1A, Pattern Based
Reason, Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
Logic
mastery makes the hard, easier. Logic
mastery leads to better, stronger and richer comprehension. Logic
mastery improves reading and writing. Logic
mastery ease learning difficulties. Logic
mastery gives a headstart. In sum, logic
mastery will develops critical thinking, improve reading and writing,
and give a firmer base for work and studies at many levels. Good luck.
After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving
liinear Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
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What may be learnt and when depends on how skills
and concepts are developed. Making the hard easier and clearer will allow
earlier & richer development of skills and concepts.
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Greatest Common Divisors and Least Common Multiples
Prime decomposition may help in the calculation of greatest common divisors
GCD
and least common multiples LCM of two to several whole numbers. The GCD is also
known as the greatest common factor. Remember
least common denominators LCD is the least common multiple of
denominators.
- To compute the Greatest common divisor, list the common prime factors and
raise each to the least multiplicities that occurs among the several whole
numbers.
- To compute the least common multiple, list all prime factors and raise
each to the greatest multiplicities that occurs among the several whole
numbers.
Then for a pair of whole numbers, their product and the product of GCD*LCM
are equal. That follows from counting the powers or multiplicities of each
prime factor.
Example:
If M = 24315273(11)4 and
N = 22305473(11)1(17)2
then the greatest common divisor is
gcd(M,N) = 2 min(4,2)3min(0,1)5min(2, 4)7min(3,3)(11)min(1,4)(17)min(0,2)
= 22305273(11)1(17)0
= 225273(11)1
and the least common multiple is
lcm(M,N) = 2 max(4,2)3max(0,1)5max(2,
4)7max(3,3)(11)max(1,4)(17)max(0,2)
= 24315473(11)4(17)2
Now min(a,b) + max(a,b) = a+ b. Therefore
gcd(M,N) * lcm(M,N) = M*N
in this example and in general.
More examples are provided by Arithmetic Videos
(RealPlayer format)
Theorem: IF the greatest common factor (divisor) of M and N is
D, then for some whole numbers A and B we have (i) M = AD and N =
DB (the latter in fact determine A and B) and we have (ii) the LCM = ADB.
Proof: See the prime decomposition method for computing the GCD and
LCD described or implied above.
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www.whyslopes.com
Number Theory
Start of Number Theory
Origins of Counting or Tallying
Adding Wholes
Multipling Wholes
Distributive Law Preamble
Distributive Law for Wholes
Consequences
More Consequences
What is a Fraction
Compound Fractions
Number Theory
Continued
Decimal Place Value
Comparison Method
Addition Method
Subtraction Methods
Multiplication Methods
Division Methods
Remainder Arithmetic I
Primes & Composites
Primes Factorization Theorem
Primes & Composites
Prime Factorization Aids
Prime Factorization Examples
Counting Whole No. Factors
Arithmetic Videos
Square Roots & Primes
Long Division Continued
Fractions & Decimals
Fractions as Decimals
1 = 0.999 Recurring
Ratio of Simple Fractions
Ratio of Decimal Fractions
Unsigned Reals Numbers
Signed Coordinates
Plane Vectors
Horizontal Vectors
Adding Vector Multiplies
Adding Signed Numbers
Multiplying Signed Numbers
Distributive Law for Reals
Real Numbers Axioms
Remainder Arithmetic II
Related Site Pages:
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck
Food for thought: Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice..
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