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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 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… ◊ ◊ ◊ ◊ ◊ 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… ◊ ◊ ◊ ◊ 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=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