Prime Factorization and a square rule
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Quick Prime Identification and Composite Number Factorization Method
If a whole number N less than 169 = 132 is not divisible by
2, 3, 5, 7 nor 11 [the primes smaller than 13] then the number is
prime. Otherwise, it - the number N - is a product of 2, 3, 5, 7 or 11
with another factor N' less than 169.
Knowledge of times tables, divisibility rules or a calculator - one
displaying results to two plus decimals, may be used to recognize
multiples of 2, 3, 5, 7 and 11. Two decimals are sufficient because the
fractions one half to one eleventh are all more than 0.01 = one
hundredth. See Efficient Square Rule Use to quickly learn more.
In learning and applying algebra exactly, one usually needs to compute
with fractions or ratios of whole numbers less than 169 or so. Learning how to efficiently use the above method for recognizing
primes and obtaining prime factorization of of whole numbers less than
169 is sufficient for most ends and purposes purposes in high school and
college mathematics and science courses. The above method is based on the
square method below.
If a whole number N less than the square of a given prime is not divisible by
all the the primes smaller than given one then the number is
prime. Otherwise, it - the number N - is a product of one smaller prime
and another factor N' less than the square of the given prime.
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This lesson focuses on obtaining prime factors and prime
decomposition of whole numbers. Prime decompositions are also known as
prime factorizations.
The the first primes 2, 3, 5, 7 or 11 are the smallest primes. A
theorem of Euclid (300 B.C.) shows there the number of primes is unlimited.
Given any finite sequence of prime numbers, their product plus one
is not a multiplie of any in the sequence. So it is prime or has
another prime as a factor. The latter observation implies one plus
the product of the first m primes has a prime factor, the m+1st or
later prime. So the list of prime numbers is un-ending - in other words
is not finite or infinite.
Theorem A. If N = AB is product of two whole numbers A and B
where A > B > 1 then A2 > N
and N2 > B.
Proof: N = AB < AA = A2 .Therefore
A2 > N. Similarly N = AB > BB =
B2 .Therefore B2 < N.
The above theorem implies if N is a product of two whole numbers, both
greater than one, than the smallest squared will be less than or
equal to N and the largest squared will be greater than or equal to N.
Now the smallest factor is a prime or it a multiple of a prime. In
either case there is prime whose square is less than or equal to the
smallest factor squared and hence less than or equal to the original
number N. Whence the next theorem holds.
Theorem B. If N is composite then N has a prime factor p with
square p2 < N.
A Square Rule Method for Identifying Primes
When an implication rule (i) If A then B holds, the contrapositive of
that rule, that is, (ii) if NOT B then NOT A, must also hold. Anything
else would contradict or be inconsistent with the original rule (i).
See the logic chapters in site Volumes 1A or 2 to learn more.
Theorem C - Contrapositive of Theorem B. If N has is not
divisible by each primes p with square p2 < N
then N is prime.
Theorem C provides the first part of square rule or method for
recognizing prime and composite numbers
The the first six primes 2, 3, 5, 7 or 11 and 13 have squares 4, 9, 25,
49, 121 and 169. That should be by your lnowledge of the times table or
decimal multiplication implies this.
Theorem C has the following consequence:
If N < 169 = 132 is not divisible by any of the primes
2, 3, 5, 7 or 11 then N is a prime number.
To check if a whole number N < 169 is a prime, is enough to check
for divisibility by 2, 3, 5, 7 or 11. The latter can be done with the
help of divisibility rules for recognizing multiples of these primes,
with the help of the 10 or 12 times table, or with the aid of a
calculator.
A Square Rule Method for Prime Factorization
Theorem: If a whole number N is composite then N has a prime factor p < then the whole
number N has a prime factor p with square p2 <
N.
Contrapositive: If a whole number N is not divisible by each
primes p with square p2 > then N is prime.
In the prime decomposition of a whole number N with the check for
divisibility of N with the smallest primes first. That will require
less arithmetic work - fewer divisions - than starting with larger
primes.
A rule of thumb, the probability that a given whole number N is
divisible by prime p is greater than the probability that N is
divisible by another prime q when p < q. Exercise - explain why.
In the prime decomposition of a whole number N, if you find q is the
smallest prime among those with square $\le$ N that divides the whole
number N, then $N = q \times M$ where M is a whole number with the
property all primes less than q do not divide M. But q will divide M if
$q^2$ divides N. Otherwise, q will not divide M. Moreover the set of
primes $p \ge q$ that have have square $\le$ M are a subset of the set
primes $p \ge q$ with with square $\le$ N. That implies fewer primes
have to be checked in the prime factorization of M. The prime
factorization of N will be given by q times the prime factorization of
M.
Remark: The square rule itself is well-known. The site innovation
consist of providing a name for it and emphasizing it as a tool to make
master of primes and prime factorization easier to learn and teach.
Exercise: Identify all multiples of 7 less than 121 that are not
also multiples of smaller primes 2, 3 and 5. The first three are
indicated above.
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Learning to do and high marks if it comes to easy is often
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