MS133

1. MS133 Chapter 4 Number Theory

2. Homework Questions

3. Test – Chapter 3 http://mcis.jsu.edu/faculty/mjohnson/ms133r3.html

4. Number Theory • Triskaidekaphobia • Palindromes 5678765 46800864 MOM BOB HANNAH RACE CAR 10/02/2001

5. Number Theory • Palindromes When was the last time the date was a palindrome before 10/02/2001? 08/31/1380

6. Number Theory • Palindromes When will the next date be a palindrome? 01/02/2010

7. Perfect Numbers • 6 is a perfect number. 6 = 1 x 6 6 = 2 x 3 6 = 1 + 2 + 3

8. Perfect Numbers • 28 is a perfect number. 28 = 1 x 28 28 = 2 x 14 28 = 4 x 7 28 = 1 + 2 + 4 + 7 + 14

9. Perfect Numbers The next perfect number is 496 6, 28, 496, 8128, . . .

10. Pythagoreans • Even – Male • Odd – Female • Marriage – 5 • Pythagorean Theorem • Irrational Numbers

11. Fermat’s Last Theorem • 1637 • an + bn = cn has no solution when n is a natural number greater than 2. • Complete proof in June 1993. (Over 350 years later) • British mathematician at Princeton University – Dr. Andrew Wiles (page 261)

12. Let a, b and c be counting numbers such that ab = c. a and b are said to be factors or divisors of c. • c is a multiple of a and c is a multipleof b. • notation: a | c and b | c (Read “a divides c” and “b divides c”)

13. 4 x 7 = 28 factor x factor = product • 4 and 7 are factors of 28 because they are used in multiplying to get a product of 28. • 28 ÷4 = 7 dividend ÷divisor = quotient • 4 and 7 are divisors of 28 because they divide into 28 evenly.

14. 28 is a multiple of 4 because it is the result of “multiplying” a counting number by 4. • 28 is a multiple of 7 because it is the result of “multiplying” a counting number by 7.

15. Divisibility Rule for 2 List the multiples of 2 (those divisible by 2) 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, . . . A number is divisible by 2 (an even number) if the digit in the one’s place is 0, 2, 4, 6, or 8. Example: 2 | 387,531,089,358 because the digit in the one’s place is 8.

16. Divisibility Rule for 5 List the multiples of 5 (those divisible by 5) 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, . . . A number is divisible by 5 if the digit in the one’s place is 0, or 5. Example: 5 | 7,678,934,620 because the digit in the one’s place is 0.

17. A number is divisible by 4 if the number formed by the last 2 digits is divisible by 4. Example: 4 | 56,720 because 4 | 20 • A number is divisible by 8 if the number formed by the last 3 digits is divisible by 8. Example: 8 | 3,567,240 because 8 | 240

18. What is the pattern? 2, 4, 8, . . . What is the pattern for the divisibility rules for 2, 4, 8, . . . ? What do you suppose the divisibility rule for 16 would be?

19. How can you tell if something is divisible by 25 (52)? (think quarters) Make a conjecture: A number is divisible by 25 (52) if

20. A number is divisible by 25 (52) if the number formed by the last 2 digits is divisible by 25. Make a conjecture for a divisibility rule for 125 (53).

21. Multiples of 3 (Numbers divisible by 3): 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, . . . Can you tell if something is divisible by 3 by looking in the one’s place? A number is divisible by 3 if the sum of the digits is divisible by 3. Example: 3|315 because 3+1+5=9 and 3|9

22. A number is divisible by 3 if the sum of the digits is divisible by 3. Example: 3|315 because 3+1+5=9 and 3|9 • A number is divisible by 9 if the sum of the digits is divisible by 9. Example: 9|783 because 7+8+3=18 and 9|18

23. A number is divisible by 7 if the number formed without the one’s place, minus twice the one’s place, is divisible by 7.

24. Example: Which of the following are divisible by 7? 511 42,975 300,846

25. A number is divisible by 11 if the sum of every other digit, minus the sum of the rest of the digits is divisible by 11.

26. Example: Which of the following is divisible by 11? 2,419,455,280 219,950,480

27. Relatively Prime Two numbers (or a group of numbers) are said to be relatively prime if they have no common factors other than 1. The only thing that will divide into both of them is 1. Examples: 20 and 21 are relatively prime. 45 and 28 are relatively prime.

28. How do you know if a number is divisible by 10? • If the number is divisible by 10, what else does it have to be divisible by? • What is the divisibility rule for 2? • What is the divisibility rule for 5?

29. If a number is divisible by 2 AND the number is divisible by 5, what has to be in the one’s place? (What is the intersection of the two rules?)

30. A number is divisible by 10 if it is divisible by 2 and 5. All other tests will be like the test for 10, a combination of the tests we have already stated.

31. How can we check to see if a number is divisible by 6?

32. How can we check to see if a number is divisible by 12? Will we check for 2 and 6 or 3 and 4? Test that on the number 18 which we know is not divisible by 12.

33. The numbers you use in combination must be RELATIVELY PRIME.

34. Use the divisibility rules to list all the factors (divisors) of 600:

35. Factors of 600 1,600 2,300 3,200 Notice that 2 is between 1 and 3 and the factor that goes with 2 is between the factor that goes with 1 and the factor that goes with 3. 300 is the only factor between 200 and 600. How will we know when we are finished?

36. Factors of 600 1,600 2,300 3,200 4,150 5,120 6,100 8,75

37. Factors of 600 1,600 10,60 2,300 3,200 4,150 5,120 6,100 8,75

38. Factors of 600 1,600 10,60 2,300 12,50 3,200 4,150 5,120 6,100 8,75

39. Factors of 600 1,600 10,60 2,300 12,50 3,200 15,40 4,150 5,120 6,100 8,75

40. Factors of 600 1,600 10,60 2,300 12,50 3,200 15,40 4,150 20,30 5,120 6,100 8,75

41. Factors of 600 1,600 10,60 2,300 12,50 3,200 15,40 4,150 20,30 5,120 24,25 6,100 8,75

42. A number is prime if it has exactly 2 factors, 1 and itself. The only way you can multiply and get the number is to multiply 1 times the number. • A number is composite if it has more than 2 factors. • 1 and 0 are neither prime nor composite.

43. Sieve of Eratosthenes 1 2 3 4 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 36 37 38 39 40 41 42 43 44 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 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

44. Sieve of Eratosthenes http://nlvm.usu.edu/en/nav/frames_asid_158_g_3_t_1.html?open=instructions

45. How do we know if a number is prime? If it is divisible by only 1 and itself. • What numbers will you check to see if they go into it? You will check to see if any prime numbers go into the number. There is no need to check any composite numbers. Any one of those would have a prime number as a factor – which you already checked. Example: If the number is not divisible by 2 it would not be divisible by any other even number.

46. How do we know if a number is prime? • If it is divisible by only 1 and itself. • You will check to see if any prime numbers go into the number. • How many prime numbers will you have to check? • If the number is 4 or more, you will have to check 2. If the number is 9 or more you will have to check 3 also. 25 or more, check 5 also. 49 or more check 7 also. 121 or more check 11 also, etc.

47. How do we know if a number is prime? • If it is divisible by only 1 and itself. • You will check to see if any prime numbers go into the number. • You will check all the prime numbers until the square of the prime number is bigger than the number in question. Do not check that one. It will not go into the number in question. • If you have not found a factor, the number is prime.

48. Composite Number • If, during that process, you find a factor, you may stop, tell what the factor is and state that the number is composite. • Remember, all it takes is one factor besides 1 and the number itself to make it be composite.

49. Tell whether each of the following is prime or composite and explain why. 151? 231? 197?

50. Write the prime factorization of 600.