DOC PREVIEW
CMU CS 15251 - lecture

This preview shows page 1-2-3-4-5-34-35-36-37-38-69-70-71-72-73 out of 73 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 73 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 73 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 73 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 73 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 73 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 73 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 73 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 73 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 73 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 73 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 73 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 73 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 73 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 73 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 73 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 73 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Slide 1Number Theory and Modular ArithmeticSlide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36What’s the pattern?Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48Slide 49Slide 50Slide 51Slide 52Slide 53Slide 54Slide 55Slide 56Slide 57Slide 58Slide 59Slide 60Slide 61Slide 62Slide 63Finally, a puzzle…why?Slide 66Slide 67Diophantine equationsNew bottles of water puzzleInvariantGeneralized bottles of waterRecall thatSlide 7315-251Great Theoretical Ideas in Computer ScienceLecture 13 (February 24, 2009)Number Theory and Modular Arithmeticp-1p1Greatest Common Divisor:GCD(x,y) = greatest k ≥ 1 s.t. k|x and k|y.Least Common Multiple:LCM(x,y) = smallest k ≥ 1 s.t. x|k and y|k.You can useMAX(a,b) + MIN(a,b) = a+bto prove the above fact…Fact:GCD(x,y) × LCM(x,y) = x × y(a mod n) means the remainder when a is divided by n. a mod n = ra = dn + r for some integer dDefinition: Modular equivalencea  b [mod n]  (a mod n) = (b mod n) n | (a-b)Written as a n b, and spoken“a and b are equivalent or congruent modulo n”31  81 [mod 2]31 2 8131  80 [mod 7]31 7 80n is an equivalence relationIn other words, it isReflexive: a n aSymmetric: (a n b) (b n a)Transitive: (a n b and b n c)  (a n c)n induces a natural partition of the integers into n “residue” classes. (“residue” = what left over = “remainder”)Define residue class [k] = the set of all integers that are congruent to k modulo n.Residue Classes Mod 3:[0] = { …, -6, -3, 0, 3, 6, ..}[1] = { …, -5, -2, 1, 4, 7, ..}[2] = { …, -4, -1, 2, 5, 8, ..} [-6] = { …, -6, -3, 0, 3, 6, ..}[7] = { …, -5, -2, 1, 4, 7, ..}[-1] = { …, -4, -1, 2, 5, 8, ..}= [0]= [1]= [2]Why do we care about these residue classes?Because we can replace any member of a residue class with another memberwhen doing addition or multiplication mod nand the answer will not changeTo calculate: 249 * 504 mod 251just do -2 * 2 = -4 = 247Fundamental lemma of plus and times mod n:If (x n y) and (a n b). Then1) x + a n y + b2) x * a n y * bProof of 2: xa = yb (mod n)(The other proof is similar…)Another Simple Fact: If (x n y) and (k|n), then: x k yExample: 10 6 16  10 3 16 Proof:A Unique Representation System Modulo n:We pick one representative from each residue class and do all our calculations using these representatives.Unsurprisingly, we use 0, 1, 2, …, n-1Unique representation system mod 3Finite set S = {0, 1, 2}+ and * defined on S:+ 0 1 20 0 1 21 1 2 02 2 0 1* 0 1 20 0 0 01 0 1 22 0 2 1Unique representation system mod 4Finite set S = {0, 1, 2, 3}+ and * defined on S:+ 0 1 2 30 0 1 2 31 1 2 3 02 2 3 0 13 3 0 1 2* 0 1 2 30 0 0 0 01 0 1 2 32 0 2 0 23 0 3 2 1NotationDefine operations +n and *n:a +n b = (a + b mod n)a *n b = (a * b mod n)Zn = {0, 1, 2, …, n-1}[“Closed”] x, y  Zn  x +n y  Zn[“Associative”] x, y, z Zn  (x +n y) +n z = x +n (y +n z)[“Commutative”]x, y  Zn  x +n y = y +n x Some properties of the operation +nSimilar properties also hold for *nUnique representation system mod 3Finite set S = {0, 1, 2}+ and * defined on S:+ 0 1 20 0 1 21 1 2 02 2 0 1* 0 1 20 0 0 01 0 1 22 0 2 1Unique representation system mod 3Finite set Z3 = {0, 1, 2}two associative, commutative operators on Z3Unique representation system mod 3Finite set Z3 = {0, 1, 2}two associative, commutative operators on Z3+ 0 1 20 0 1 21 1 2 02 2 0 1* 0 1 20 0 0 01 0 1 22 0 2 1Unique representation system mod 2Finite set Z2 = {0, 1}two associative, commutative operators on Z2+20 10 0 11 1 0*20 10 0 01 0 1XORANDZ5 = {0,1,2,3,4}+ 0 1 2 3 40 0 1 2 3 41 1 2 3 4 02 2 3 4 0 13 3 4 0 1 24 4 0 1 2 3* 0 1 2 3 40 0 0 0 0 01 0 1 2 32 03 0 3 1 44 0 4 3 2Z6 = {0,1,2,3,4,5}+ 0 1 2 3 4 50 0 1 2 3 4 51 1 2 3 4 5 02 2 3 4 5 0 13 3 4 5 0 1 24 4 5 0 1 2 35 5 0 1 2 3 4* 0 1 2 3 4 50 0 0 0 0 01 0 1 2 3 42 0 2 4 0 23 04 0 4 2 0 45 0 5 4 3 2For addition tables, rows and columnsalways are a permutation of Zn+ 0 1 2 3 4 50 0 1 2 3 4 51 1 2 3 4 5 02 2 3 4 5 0 13 3 4 5 0 1 24 4 5 0 1 2 35 5 0 1 2 3 4+ 0 1 2 3 40 0 1 2 3 41 1 2 3 4 02 2 3 4 0 13 3 4 0 1 24 4 0 1 2 3For multiplication, some rows and columnsare permutation of Zn, while others aren’t…* 0 1 2 3 40 0 0 0 0 01 0 1 2 3 42 0 2 4 1 33 0 3 1 4 24 0 4 3 2 1* 0 1 2 3 4 50 0 0 0 0 0 01 0 1 2 3 4 52 0 2 4 0 2 43 0 3 0 3 0 34 0 4 2 0 4 25 0 5 4 3 2 1what’s happening here?For addition, the permutation propertymeans you can solve, say,+ 0 1 2 3 4 50 0 1 2 3 4 51 1 2 3 4 5 02 2 3 4 5 0 13 3 4 5 0 1 24 4 5 0 1 2 35 5 0 1 2 3 44 + ___ = 1 (mod 6)Subtraction mod n is well-definedEach row has a 0,hence –a is that elementsuch that a + (-a) = 0 a – b = a + (-b)4 + ___ = x (mod 6) for any x in Z6For multiplication, if a row has a permutationyou can solve, say,5 * ___ = 4 (mod 6)* 0 1 2 3 4 50 0 0 0 0 0 01 0 1 2 3 4 52 0 2 4 0 2 43 0 3 0 3 0 34 0 4 2 0 4 25 0 5 4 3 2 1or, 5 * ___ = 1 (mod 6)But if the row does not …


View Full Document

CMU CS 15251 - lecture

Documents in this Course
lecture

lecture

66 pages

lecture

lecture

79 pages

lecture

lecture

111 pages

lecture

lecture

85 pages

lecture17

lecture17

64 pages

Lecture

Lecture

85 pages

Lecture

Lecture

71 pages

Lecture

Lecture

70 pages

Lecture

Lecture

11 pages

Lecture

Lecture

45 pages

Lecture

Lecture

50 pages

Lecture

Lecture

93 pages

Lecture

Lecture

93 pages

Lecture

Lecture

35 pages

Lecture

Lecture

98 pages

Lecture

Lecture

74 pages

Lecture

Lecture

13 pages

Lecture

Lecture

15 pages

Lecture

Lecture

66 pages

Lecture

Lecture

82 pages

Lecture

Lecture

15 pages

Lecture

Lecture

47 pages

Lecture

Lecture

69 pages

Lecture

Lecture

13 pages

Lecture

Lecture

67 pages

Lecture

Lecture

68 pages

Lecture

Lecture

69 pages

lecture03

lecture03

44 pages

Lecture

Lecture

69 pages

Lecture

Lecture

68 pages

Lecture

Lecture

55 pages

Lecture

Lecture

79 pages

Lecture

Lecture

85 pages

Lecture

Lecture

87 pages

Lecture

Lecture

85 pages

Lecture

Lecture

103 pages

Lecture

Lecture

9 pages

Lecture

Lecture

83 pages

Lecture

Lecture

8 pages

lecture03

lecture03

68 pages

lecture24

lecture24

78 pages

lecture03

lecture03

72 pages

Thales

Thales

129 pages

lecture13

lecture13

81 pages

Lecture

Lecture

64 pages

lecture01

lecture01

59 pages

lecture11

lecture11

105 pages

Lecture

Lecture

89 pages

Lecture

Lecture

74 pages

lecture25

lecture25

57 pages

Lecture

Lecture

99 pages

lecture

lecture

50 pages

lecture

lecture

14 pages

Lecture

Lecture

78 pages

lecture

lecture

8 pages

Lecture

Lecture

98 pages

lecture

lecture

83 pages

lecture23

lecture23

88 pages

lecture

lecture

64 pages

lecture

lecture

72 pages

Lecture

Lecture

88 pages

lecture

lecture

79 pages

Lecture

Lecture

60 pages

lecture

lecture

74 pages

lecture19

lecture19

72 pages

lecture25

lecture25

86 pages

lecture

lecture

13 pages

lecture17

lecture17

79 pages

lecture

lecture

91 pages

lecture

lecture

78 pages

Lecture

Lecture

11 pages

Lecture

Lecture

54 pages

lecture

lecture

72 pages

lecture

lecture

119 pages

lecture

lecture

167 pages

lecture

lecture

73 pages

lecture

lecture

83 pages

lecture

lecture

49 pages

lecture

lecture

16 pages

lecture

lecture

67 pages

lecture

lecture

81 pages

lecture

lecture

72 pages

lecture

lecture

57 pages

lecture16

lecture16

82 pages

lecture21

lecture21

46 pages

Lecture

Lecture

92 pages

Lecture

Lecture

14 pages

Lecture

Lecture

49 pages

Lecture

Lecture

132 pages

Lecture

Lecture

101 pages

Lecture

Lecture

98 pages

Lecture

Lecture

59 pages

Lecture

Lecture

64 pages

Lecture

Lecture

106 pages

Lecture

Lecture

70 pages

Lecture

Lecture

80 pages

Lecture

Lecture

76 pages

Lecture

Lecture

91 pages

Lecture

Lecture

112 pages

Lecture

Lecture

91 pages

Lecture

Lecture

10 pages

Lecture

Lecture

39 pages

Lecture

Lecture

79 pages

Lecture

Lecture

74 pages

Lecture

Lecture

44 pages

Lecture

Lecture

39 pages

Lecture

Lecture

99 pages

Lecture

Lecture

44 pages

Lecture

Lecture

59 pages

Lecture

Lecture

36 pages

lecture17

lecture17

36 pages

lecture

lecture

71 pages

lecture

lecture

79 pages

lecture

lecture

12 pages

lecture

lecture

43 pages

lecture

lecture

87 pages

lecture

lecture

35 pages

lecture03

lecture03

23 pages

lecture

lecture

68 pages

lecture

lecture

74 pages

lecture

lecture

21 pages

lecture

lecture

79 pages

lecture

lecture

15 pages

lecture

lecture

83 pages

lecture

lecture

13 pages

Lecture

Lecture

53 pages

lecture

lecture

55 pages

lecture

lecture

49 pages

lecture

lecture

10 pages

lecture

lecture

70 pages

lecture

lecture

12 pages

Lecture

Lecture

105 pages

Lecture

Lecture

9 pages

Lecture

Lecture

72 pages

Lecture

Lecture

66 pages

Lecture

Lecture

54 pages

Lecture

Lecture

98 pages

Lecture

Lecture

57 pages

Lecture

Lecture

75 pages

Lecture

Lecture

48 pages

lecture

lecture

53 pages

Lecture

Lecture

72 pages

Lecture

Lecture

53 pages

Lecture

Lecture

84 pages

Lecture

Lecture

55 pages

Lecture

Lecture

15 pages

Lecture

Lecture

6 pages

Lecture

Lecture

38 pages

Lecture

Lecture

71 pages

Lecture

Lecture

110 pages

Lecture

Lecture

70 pages

lecture

lecture

48 pages

lecture

lecture

76 pages

lecture

lecture

48 pages

lecture

lecture

52 pages

lecture

lecture

43 pages

lecture

lecture

81 pages

lecture

lecture

82 pages

lecture

lecture

83 pages

lecture

lecture

64 pages

lecture

lecture

71 pages

lecture

lecture

65 pages

lecture

lecture

56 pages

lecture

lecture

12 pages

lecture

lecture

66 pages

lecture

lecture

50 pages

lecture

lecture

86 pages

lecture

lecture

70 pages

Lecture

Lecture

74 pages

Lecture

Lecture

54 pages

Lecture

Lecture

90 pages

lecture

lecture

78 pages

lecture

lecture

87 pages

Lecture

Lecture

55 pages

Lecture

Lecture

12 pages

lecture21

lecture21

66 pages

Lecture

Lecture

11 pages

lecture

lecture

83 pages

Lecture

Lecture

53 pages

Lecture

Lecture

69 pages

Lecture

Lecture

12 pages

lecture04

lecture04

97 pages

Lecture

Lecture

14 pages

lecture

lecture

75 pages

Lecture

Lecture

74 pages

graphs2

graphs2

8 pages

lecture

lecture

82 pages

Lecture

Lecture

8 pages

lecture

lecture

47 pages

lecture

lecture

91 pages

lecture

lecture

76 pages

lecture

lecture

73 pages

lecture

lecture

10 pages

lecture

lecture

63 pages

lecture

lecture

91 pages

lecture

lecture

79 pages

lecture

lecture

9 pages

lecture

lecture

70 pages

lecture

lecture

86 pages

lecture

lecture

102 pages

lecture

lecture

145 pages

lecture

lecture

91 pages

Lecture

Lecture

87 pages

lecture

lecture

87 pages

Notes

Notes

19 pages

Lecture

Lecture

50 pages

Lecture

Lecture

13 pages

Lecture

Lecture

97 pages

Lecture

Lecture

98 pages

Lecture

Lecture

83 pages

Lecture

Lecture

77 pages

Lecture

Lecture

102 pages

Lecture

Lecture

63 pages

Lecture

Lecture

104 pages

lecture

lecture

41 pages

lecture

lecture

14 pages

Lecture

Lecture

87 pages

Lecture

Lecture

94 pages

lecture

lecture

9 pages

Lecture

Lecture

96 pages

Lecture

Lecture

72 pages

Lecture

Lecture

35 pages

Lecture

Lecture

77 pages

Lecture

Lecture

98 pages

Lecture

Lecture

48 pages

Lecture

Lecture

66 pages

Lecture

Lecture

53 pages

lecture18

lecture18

101 pages

Lecture

Lecture

10 pages

Lecture

Lecture

70 pages

Lecture

Lecture

12 pages

Lecture

Lecture

74 pages

graphs

graphs

10 pages

Lecture

Lecture

62 pages

Lecture

Lecture

11 pages

Lecture

Lecture

71 pages

Lecture

Lecture

42 pages

lecture15

lecture15

72 pages

Lecture

Lecture

82 pages

Load more
Download lecture
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view lecture and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view lecture 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?