Factorization and Primality
Pure mathematics is, in its way, the poetry of
logical ideas. ~Albert Einstein
What is Primality?
◊ A prime number can only
be divided by itself or 1
without a remainder.
◊ Some examples include
7, 11, and 13.
◊ Can you think of any
other prime numbers?
◊ A number which
exhibits primality is
considered a prime
number.
History of Primality
◊ A primality test is a shortcut to find
prime numbers.
◊ The earliest known method was
developed around 284 BC by a
Greek mathematician known as
Eratosthenes.
◊ This method is known as the
Sieve of Eratosthenes.
◊ Using this method with the fastest computers to verify
a number’s primality would take longer than the life
expectancy of our sun! (5 billion years)
Sieve of Eratosthenes
◊ For primes less than 30 and greater than 1…
A Quick Exercise
◊ On the handout, using the Sieve of Eratosthenes, find the
prime numbers on these intervals…
a) 30 < x < 40 … (√40 < 7)
b) 5 < x < 35 … (√35 < 6)
c) 45 < x < 75 … (√75 < 9)
30 < x < 40
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First write out all the numbers…
[30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]
Always begin by erasing all numbers that are multiples of 2…
Now we have … [31, 33, 35, 37, 39]
Next, erase all numbers that are multiples of 3…
Now we have … [31, 35, 37]
Erase all numbers that are multiples of 5…
Finally, we have … [31, 37]
◊ The condition states that we stop at this multiple because √40 < 7
Note: (since the numbers 4 and 6 are not prime we need not worry about their multiples)
5 < x < 35
◊ First write out all the numbers…
◊ [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30, 31, 32, 33, 34, 35]
◊ Always begin by erasing all numbers that are multiples of 2…
◊ Now we have … [5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35]
◊ Next, erase all numbers that are multiples of 3…
◊ Now we have … [5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35]
◊ Erase all numbers that are multiples of 5…
◊ Finally, we have … [5, 7, 11, 13, 17, 19, 23, 29, 31]
◊ The condition states that we stop at this multiple because √35 < 6
Note: (since the number 4 is not prime we need not worry about its multiples)
45 < x < 75
◊ First write out all the numbers…
◊ [45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64,
65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75]
◊ Always begin by erasing all numbers that are multiples of 2…
◊ Now we have … [45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75]
◊ Erase all numbers that are multiples of 3…
◊ Now we have … [47, 49, 53, 55, 59, 61, 65, 67, 71, 73]
◊ Erase all numbers that are multiples of 5…
◊ Now we have … [47, 49, 53, 59, 61, 67, 71, 73]
◊ Erase all numbers that are multiples of 7…
◊ Finally we have … [47, 53, 59, 61, 67, 71, 73]
◊ The condition states that we stop at this multiple because √75 < 9
Note: (since the numbers 4, 6, and 8 are not prime we need not worry about their multiples)
History of Primality (cont.)
◊ Fibonacci, while being
tutored by an Arab scholar
in Africa, developed the
first deterministic algorithm
for primality testing.
◊ This was important
because this type of
algorithm is simpler than
the algorithms used before,
known as randomized
algorithms.
What is Factorization?
◊ Factorization is the resolution of a unit into factors.
◊ A factor can be numbers or algebraic expressions that
when multiplied together give the initial unit.
◊ An example of this is the unit 24; the factors of 24 are
2*2*2*3
Factoring Tricks
◊ 2 is a factor of any even number.
◊ 3 is a factor of any number whose sum of individual
numbers are divisible by 3.
◊ 4 is a factor of any number whose last two digits are
divisible by 4.
◊ 5 is a factor of any number ending in a 5 or 0.
◊ 6 is a factor of any number that is divisible by 2 and 3.
◊ 8 is a factor of any number whose last 3 digits are
divisible by 8.
◊ 9 is a factor of any number whose sum of individual
numbers are divisible by 9.
◊ 10 is a factor of any number ending in a 0.
Factoring Tricks (cont.)
◊ 7 is a factor of a number if the last digit, multiplied
by 2 and subtracted from the remaining digits, is
divisible by 7…
◊ Take the number 1,484 for example…
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The last digit is 4 so …
4*2 = 8
The remaining digits are 148 so …
148 – 8 = 140
The number 140 is divisible by 7! (140 / 7 = 20)
◊ Therefore 7 is a factor of 1,484. To check …
◊ 1,484 / 7 =212
Factoring Tricks (cont.)
◊ 11 is a factor of a number if the sum of every other
number minus the sum of the remaining individual
numbers is divisible by 11.
◊ Take the number 1,045 for example…
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First, sum every other number … (1+4 = 5)
Next sum the remaining numbers … (0+5 = 5)
Subtract these two sums … (5-5 = 0)
The number 0 is divisible by 11! (0/11 = 0)
◊ Therefore 11 is a factor of 1,045. To check …
◊ 1,045/11 = 95
Finding the GCF
Using the factoring tricks, find the GCF of 3,960 and 2,520:
◊ List the prime factors of each number and circle the
factors common to both:
◊ 3,960 = 11 X 3 X 3 X 2 X 2 X 2 X 5
◊ 2,520 = 7 X 3 X 3 X 2 X 2 X 2 X 5
◊ Since 3 X 3 X 2 X 2 X 2 X 5 = 360 are common in both
numbers, this is the GCF.
Applications in Real World
◊ Mathematicians, like Eratosthenes and Fibonacci,
paved the way for many current uses of prime
numbers and factorization in today's world.
◊ In computer science, data
compression is the process
of encoding information
using fewer bits than
normally used.
◊ The ZIP file format is a
good example of data
compression. ZIP files
store many files in a single
output file.
Applications in Real World
◊ Data transmission is another example of how
primality and factorization is used today.
◊ Specifically, the area of protocol and
handshaking in data transmission is where
these basic concepts are utilized.
◊ A protocol is an agreed-upon format,
defined by a set of rules, for transmitting
data between two devices like a computer
and printer for example.
◊ Handshaking occurs when two devices
send several messages back and forth,
enabling them to agree on a
communications protocol.
Works Cited
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homework.syosset.k12.ny.us/teachers/jconnoll/
www.quotegarden.com/math.html
www.dictionary.com
www.childrensmuseum.org
www.jstor.org/view/0025570x/sp050003/05x0111g/1?frame=no
frame&userID=a55b1ccb@tamu.edu/01c0a8346a00501cdeb71
&dpi=3&config=jstor
◊ faculty.evansville.edu
◊ www.energysafety.govt.nz
◊ http://www.mathdoesntsuck.com/downloadable/Divisibilitytricks-7and11-rev.pdf