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

◊ 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 Note: (since the numbers 4 and 6 are not prime we need not worry about their multiples)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)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 Note: (since the numbers 4, 6, and 8 are not prime we need not worry about their multiples)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

◊ 22 is a factor of any even number.◊ 33 is a factor of any number whose sum of individual

numbers are divisible by 3. ◊ 44 is a factor of any number whose last two digits are

divisible by 4.◊ 55 is a factor of any number ending in a 5 or 0.◊ 66 is a factor of any number that is divisible by 2 and

3.◊ 88 is a factor of any number whose last 3 digits are

divisible by 8.◊ 99 is a factor of any number whose sum of individual

numbers are divisible by 9.◊ 1010 is a factor of any number ending in a 0.

Factoring Tricks (cont.)

◊ 77 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… ◊ 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.)

◊ 1111 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…◊ 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

◊ 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=noframe&userID=a55b1ccb@tamu.edu/01c0a8346a00501cdeb71&dpi=3&config=jstor

◊ faculty.evansville.edu◊ www.energysafety.govt.nz◊ http://www.mathdoesntsuck.com/downloadable/

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