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Integers, Division Primes, Greatest Common Divisors, Modular Arithmetic

Integers, Division Primes, Greatest Common Divisors, Modular Arithmetic. Sections 4.1 Tuesday 3 June. 1. Division. Definition : If a and b are integers with a ≠ 0 , then a divides b if there exists an integer c such that b = ac .

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Integers, Division Primes, Greatest Common Divisors, Modular Arithmetic

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  1. Integers, DivisionPrimes, Greatest Common Divisors, Modular Arithmetic Sections 4.1 Tuesday 3 June INTEGERS, DIVISION, PRIMES 1

  2. Division • Definition: If a and b are integers with a ≠ 0, then adividesb if there exists an integer c such that b = ac. • When a divides b we say that a is a factorof b, or a is a divisor of b and that b is a multiple of a. • The notationa | b denotes that a divides b. • If a | b, then b/a is an integer. • If a does not divide b, we writea∤b. • Exercise: Which of the following are true? • 3 | 7 • 3 | 12 • 5 ∤15 INTEGERS, DIVISION, PRIMES

  3. Properties of Divisibility • Theorem 1: Let a, b, and c be integers, where a ≠0. • If a | b and a | c, then a | (b + c); • If a | b, then a | (bc) for all integers c; • If a | b and b | c, then a | c. • Proof: • Suppose a | b and a | c. • Then thereare integers s and t with b = as and c = at. • Hence, b + c = as + at = a(s + t). • It follows thata | (b + c). • Hence, if a | b and a | c, then a | (b + c). • (ii), (iii) : Exercise (the proofs are similar to the proof above). INTEGERS, DIVISION, PRIMES

  4. Properties of Divisibility Theorem 1: Let a, b, and c be integers, where a ≠0. • If a | b and a | c, then a | (b + c); • If a | b, then a | (bc) for all integers c; • If a | b and b | c, then a | c. Corollary: If a, b, and c be integers such that a ≠ 0 and a | b and a | c, then a | (mb + nc) whenever m and n are integers. Exercise: Show how this Corollary follows from Theorem 1. INTEGERS, DIVISION, PRIMES

  5. Properties of Divisibility • Lemma: If n is a positive integer and a is a positive factor of n, then 1 ≤ a ≤ n. • Proof: Assume n is a positive integer and a is a positive factor of n. Then n = ab, for some integer b. Moreover, b must be positive since both n and a are. Hence b ≥ 1. From this inequality we get: n = ab ≥ a ∙1 = a. Additionally, the assumption that a is positive implies a ≥ 1. Thus, 1 ≤ a ≤ n. INTEGERS, DIVISION, PRIMES

  6. Division Algorithm • Division Algorithm (Theorem): If a is an integer and d is a positive integer, then there are unique integers q and r, with 0 ≤ r < d, such that a = dq + r. • d is called the divisor. • a is called the dividend. • q is called the quotient. • r is called the remainder. • Exercises: Find the following: • 39 div 15 and 39 mod 15 • 45 div 15 and 45 mod 15 • -20 div 15 and -20 mod 15 • Definitions of div and mod: • q = a div d • r = a mod d INTEGERS, DIVISION, PRIMES

  7. Congruence Relation • Definition: If a and b are integers and m is a positive integer, thena is congruent to b modulo m iff m | (a – b). • We write a ≡ b (mod m) to say that a is congruent to b modulo m. • Two integers are congruent mod m if and only if they have the same remainder when divided by m. • We write a ≢ b (mod m) to say a is not congruent to b modulo m. • Exercise: Which of the following are true? • 17 ≡ 5 (mod 6) • 24 ≡ 14 (mod 6) • 24 ≡ −14 (mod 6) • −15 ≡ −15 (mod 6) INTEGERS, DIVISION, PRIMES

  8. More on Congruences • Theorem 4: Let m be a positive integer. The integers a and b are congruent modulo m if and only if there is an integer k such that a = b + km. • Proof: • (=>) Assume a ≡ b (mod m) Then m | (a – b). Hence, there is an integer k such that (a – b) = km. But then a = b + km. • (<=) Assume there is an integer k such that a = b + km, Then km = a – b. Hence, m | (a – b). Thus, a ≡ b (mod m). INTEGERS, DIVISION, PRIMES

  9. Notation Hazard: (mod m) v.s. mod • The “mod” in a ≡ b (mod m) and amodm = b are different. • a ≡ b (mod m) is a relation on the set of integers. • In amodm, the notation mod denotes a binary operation (function). • Theorem 3: Let a and b be integers, and let m be a positive integer. Then a ≡ b (mod m) if and only if amodm = bmodm. (Proof in the exercises) INTEGERS, DIVISION, PRIMES

  10. Congruences of Sums and Products • Theorem 5: Let m be a positive integer. If a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m) and ac ≡ bd (mod m). • Proof: • Assume a ≡ b (mod m) and c ≡ d (mod m). • Then, by Theorem 4, there are integers s and t with b = a + sm and d = c + tm. • Therefore, b + d = (a + sm) + (c + tm) = (a + c) + m(s + t) and bd = (a + sm) (c + tm) = ac + m(at + cs + stm). • Hence, by Theorem 4, a + c ≡ b + d (mod m) and ac ≡ bd (mod m). • Example: Because 7 ≡ 2 (mod 5) and 11 ≡ 1 (mod 5) , it follows that 18 = 3 (mod 5) and 77 = 2 (mod 5). INTEGERS, DIVISION, PRIMES

  11. Algebraic Manipulation of Congruences • Multiplying both sides of a valid congruence by an integer preserves validity. If a ≡b (mod m) is true then c ∙ a ≡c ∙ b (mod m) is true, where c is any integer. Proof: Theorem 5 since c = c (mod m). • Adding an integer to both sides of a valid congruence preserves validity. If a ≡b (mod m) is true then c + a ≡c + b (mod m) is true, where c is any integer. Proof: Theorem 5 since c = c (mod m). • Dividing both sides of a valid congruence by an integer does not always produce a valid congruence. Example: The congruence 14 ≡ 8 (mod 6) holds. But dividing both sides by 2 does not produce a valid congruence: 14/2 = 7 and 8/2 = 4, but 7≢4 (mod 6). INTEGERS, DIVISION, PRIMES

  12. Computing mod for Products and Sums • Corollary: Let m be a positive integer and let a and b be integers. Then the following are true: • (a + b) modm = ((amodm) + (bmodm)) modm • abmodm = ((a modm) (bmodm)) modm. • Allows you to compute the remainder of the product (sum) of two integers when divided by m from the remainders when each is divided by m. • Exercise: Use this corollary to find the following • 240009 mod 12 • ((39)(53)) mod 11 INTEGERS, DIVISION, PRIMES

  13. Arithmetic Modulo m • Definitions: LetZm be the set of nonnegative integers less than m: Zm = {0,1, …., m − 1} • The operation +m is defined as a +mb = (a + b) modm. This is addition modulo m. • The operation ∙m is defined as a ∙m b = (a + b) modm. This is multiplication modulo m. • Using these operations is called doing arithmetic modulo m. • Example: Find 7 +11 9 and 7 ∙11 9. • Solution: Using the definitions above: • 7 +11 9 = (7 + 9) mod 11 = 16 mod 11 = 5 • 7 ∙11 9 = (7 ∙ 9) mod 11 = 63 mod 11 = 8 INTEGERS, DIVISION, PRIMES

  14. Arithmetic Modulo m • The operations +m and ∙m satisfy many of the same properties as ordinary addition and multiplication. • Closure: If a and b belong toZm , then a +m b and a ∙m b belong to Zm . • Associativity: If a, b, and c belong to Zm , then (a +m b) +m c = a +m (b +m c) and (a ∙m b) ∙m c = a ∙m (b ∙mc). • Commutativity: If a and b belong to Zm , then a +m b = b +m a and a ∙m b = b ∙m a. • Identity elements: If a belongs to Zm , then a +m 0 = a and a ∙m1 = a. continued→ INTEGERS, DIVISION, PRIMES

  15. Arithmetic Modulo m • Additive inverses: If a ≠ 0 belongs to Zm , then m − a is the additive inverse of a modulo m and 0 is its own additive inverse. • a +m (m− a ) = 0 and 0 +m 0 = 0 • Distributivity: If a, b, and c belong to Zm , then • a∙m (b +m c) = (a∙m b) +m (a ∙m c) and (a+m b)∙m c = (a ∙m c) +m (b∙m c). • Proofs are exercises. • Multiplicative inverses have not been included since they do not always exist. For example, there is no multiplicative inverse of 2 modulo 6. • (optional) Using the terminology of abstract algebra, Zm with +m is a commutative group and Zm with +m and ∙m is a commutative ring. INTEGERS, DIVISION, PRIMES

  16. Primes • Definition: A positive integer p greater than 1 is prime if the only positive factors of p are 1 and p. • A positive integer that is greater than 1 and is not prime is called composite. • Exercise: Classify each of the following numbers as prime, composite, or neither. • 11 • -11 • 93 • 1 • 2 INTEGERS, DIVISION, PRIMES

  17. The Fundamental Theorem of Arithmetic • Theorem: Every positive integer greater than 1 can be written as a prime or as the product of two or more primes; this representation is unique if the prime factors are written in order of non-decreasing size. • Examples: • 100 = 2 ∙ 2 ∙ 5 ∙ 5 = 22 ∙ 52 • 641 = 641 • 999 = 3 ∙ 3 ∙ 3 ∙ 37 = 33 ∙ 37 • 1024 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 = 210 INTEGERS, DIVISION, PRIMES

  18. The Sieve of Erastosthenes Erastothenes (276-194 B.C.) • A method for finding all primes that do not exceed a given positive integer, n. • List all of the integers from 2 to n in increasing order. 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 INTEGERS, DIVISION, PRIMES

  19. The Sieve of Erastosthenes Erastothenes (276-194 B.C.) • A method for finding all primes that do not exceed a given positive integer, n. • List all of the integers from 2 to n in increasing order. • Mark the first unmarked element of the list as “prime”. 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 INTEGERS, DIVISION, PRIMES

  20. The Sieve of Erastosthenes Erastothenes (276-194 B.C.) • A method for finding all primes that do not exceed a given positive integer, n. • List all of the integers from 2 to n in increasing order. • Mark the first unmarked element of the list as “prime”. • Delete all the unmarked integers that are divisible by the last element that was marked as “prime”. • Repeat the previous two steps until only marked integers are left. 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 INTEGERS, DIVISION, PRIMES

  21. The Sieve of Erastosthenes Erastothenes (276-194 B.C.) • A method for finding all primes that do not exceed a given positive integer, n. • List all of the integers from 2 to n in increasing order. • Mark the first unmarked element of the list as “prime”. • Delete all the unmarked integers that are divisible by the last element that was marked as “prime”. • Repeat the previous two steps until only marked integers are left. 2 34 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 INTEGERS, DIVISION, PRIMES

  22. The Sieve of Erastosthenes Erastothenes (276-194 B.C.) • A method for finding all primes that do not exceed a given positive integer, n. • List all of the integers from 2 to n in increasing order. • Mark the first unmarked element of the list as “prime”. • Delete all the unmarked integers that are divisible by the last element that was marked as “prime”. • Repeat the previous two steps until only marked integers are left. 2 34 5 6 7 8 910 11 12 13 1415 16 17 18 19 20 2122 23 24 25 26 2728 29 30 31 32 3334 35 36 37 383940 41 42 43 4445 46 47 48 49 5051 52 53 54 55 565758 59 60 61 62 6364 65 66 67 686970 71 72 73 74 7576 77 78 79 808182 83 84 85 86 87 88 89 90 INTEGERS, DIVISION, PRIMES

  23. The Sieve of Erastosthenes Erastothenes (276-194 B.C.) • A method for finding all primes that do not exceed a given positive integer, n. • List all of the integers from 2 to n in increasing order. • Mark the first unmarked element of the list as “prime”. • Delete all the unmarked integers that are divisible by the last element that was marked as “prime”. • Repeat the previous two steps until only marked integers are left. 2 34 56 7 8 910 11 12 13 1415 16 17 18 19 20 2122 23 24 25 26 2728 29 30 31 32 3334 35 36 37 383940 41 42 43 4445 46 47 48 49 5051 52 53 54 55 565758 59 60 61 62 6364 65 66 67 686970 71 72 73 74 7576 77 78 79 808182 83 84 85 86 87 88 89 90 INTEGERS, DIVISION, PRIMES

  24. The Sieve of Erastosthenes Erastothenes (276-194 B.C.) • A method for finding all primes that do not exceed a given positive integer, n. • List all of the integers from 2 to n in increasing order. • Mark the first unmarked element of the list as “prime”. • Delete all the unmarked integers that are divisible by the last element that was marked as “prime”. • Repeat the previous two steps until only marked integers are left. 2 34 56 7 8 910 11 12 13 1415 16 17 18 19 20 2122 23 242526 27 28 29 30 31 32 3334 3536 37 383940 41 42 43 4445 46 47 48 49 5051 52 53 54 55565758 59 60 61 62 63646566 67 686970 71 72 73 74 7576 77 78 79 808182 83 848586 87 88 89 90 INTEGERS, DIVISION, PRIMES

  25. The Sieve of Erastosthenes Erastothenes (276-194 B.C.) • A method for finding all primes that do not exceed a given positive integer, n. • List all of the integers from 2 to n in increasing order. • Mark the first unmarked element of the list as “prime”. • Delete all the unmarked integers that are divisible by the last element that was marked as “prime”. • Repeat the previous two steps until only marked integers are left. 2 34 567 8 910 11 12 13 1415 16 17 18 19 20 2122 23 242526 2728 2930 31 32 3334 3536 37 383940 41 42 43 4445 46 47 48 49 5051 52 53 54 55565758 59 60 61 62 63646566 67 686970 71 72 73 74 7576 77 78 79 808182 83 848586 87 88 89 90 INTEGERS, DIVISION, PRIMES

  26. The Sieve of Erastosthenes Erastothenes (276-194 B.C.) • A method for finding all primes that do not exceed a given positive integer, n. • List all of the integers from 2 to n in increasing order. • Mark the first unmarked element of the list as “prime”. • Delete all the unmarked integers that are divisible by the last element that was marked as “prime”. • Repeat the previous two steps until only marked integers are left. 2 34 567 8 910 11 12 13 1415 16 17 18 19 20 2122 23 242526 2728 29 30 31 32 3334 3536 37 383940 41 42 43 4445 46 47 48 495051 52 53 54 55565758 59 60 61 62 63646566 67 686970 71 72 73 74 7576 77 78 79 808182 83 848586 87 88 89 90 INTEGERS, DIVISION, PRIMES

  27. The Sieve of Erastosthenes Erastothenes (276-194 B.C.) • A method for finding all primes that do not exceed a given positive integer, n. • List all of the integers from 2 to n in increasing order. • Mark the first unmarked element of the list as “prime”. • Delete all the unmarked integers that are divisible by the last element that was marked as “prime”. • Repeat the previous two steps until only marked integers are left. 2 34 567 8 910 1112 13 1415 16 17 18 19 20 2122 23 242526 2728 29 30 31 32 3334 3536 37 383940 41 42 43 4445 46 47 48 495051 52 53 54 55565758 59 60 61 62 63646566 67 686970 71 72 73 74 7576 77 78 79 808182 83 848586 87 88 89 90 INTEGERS, DIVISION, PRIMES

  28. The Sieve of Erastosthenes Erastothenes (276-194 B.C.) • A method for finding all primes that do not exceed a given positive integer, n. • List all of the integers from 2 to n in increasing order. • Mark the first unmarked element of the list as “prime”. • Delete all the unmarked integers that are divisible by the last element that was marked as “prime”. • Repeat the previous two steps until only marked integers are left. 2 34 567 8 910 1112131415 16 17 18 19 20 2122 23 242526 2728 29 30 31 32 3334 3536 37 383940 41 42 43 4445 46 47 48 495051 52 53 54 55565758 59 60 61 62 63646566 67 686970 71 72 73 74 7576 77 78 79 808182 83 848586 87 88 89 90 And so on, until … INTEGERS, DIVISION, PRIMES

  29. The Sieve of Erastosthenes Erastothenes (276-194 B.C.) • A method for finding all primes that do not exceed a given positive integer, n. • List all of the integers from 2 to n in increasing order. • Mark the first unmarked element of the list as “prime”. • Delete all the unmarked integers that are divisible by the last element that was marked as “prime”. • Repeat the previous two steps until only marked integers are left. 2 34 567 8 910 1112131415 1617181920 212223242526 27282930 3132 3334 353637 383940 4142434445 46 4748 495051 525354 55565758596061 62 6364656667 68697071727374 7576 77 78 79808182 83848586 87 888990 INTEGERS, DIVISION, PRIMES

  30. The Sieve of Erastosthenes Erastothenes (276-194 B.C.) • When could you stop marking numbers and just say “the remaining unmarked integers are all prime”? • Theorem 2: If n is a composite integer, then n has a prime divisor less than or equal to √n Proof: Assume n is composite. Then n = ab, for some integers, a and b, both greater than 1. We show by contradiction that either a ≤ √n or b ≤ √n. Assume a > √n and b > √n. (*) Then ab > n, which contradicts the choice of a and b. Hence, the assumption (*) must be false. QED. • This theorem justifies stopping at √n INTEGERS, DIVISION, PRIMES

  31. Infinitude of Primes Euclid (325 B.C. – 265 B.C.) • Theorem: There are infinitely many primes. (Euclid) • Proof: • Assume that there are only n primes: p1, p2, …, pn • Let q = p1p2∙∙∙pn + 1 • Either q is prime or it is a product of primes (Fund. Thm. Arith.). • But none of the primes pj divides q since if pj | q, then pj dividesq − p1p2∙∙∙ pn = 1 and 1 has no prime factors. • As these are the only primes, q must be prime. • But q > pj, for all the pj. • So, contrary to our starting assumption, there are at least n + 1 primes. • Consequently, there are infinitely many primes. This proof was given by Euclid The Elements. The proof is considered to be one of the most beautiful in all mathematics. It is the first proof in The Book, inspired by the famous mathematician Paul Erdős’ imagined collection of perfect proofs maintained by God. Paul Erdős (1913-1996) INTEGERS, DIVISION, PRIMES

  32. Generating Primes • Finding large primes with hundreds of digits is important in cryptography. • So far, no one has found a closed formula that always produces primes. • f(n) = n2 − n + 41 is prime for all integers 1,2,…, 40. But f(41) = 412 is not prime. • More generally, there is no polynomial with integer coefficients such that f(n) is prime for all positive integers n. • Fortunately, we can generate large integers which are almost certainly prime. INTEGERS, DIVISION, PRIMES

  33. Conjectures about Primes Many conjectures about primes are unresolved, including: • Goldbach’s Conjecture: Every even integer n, n > 2, is the sum of two primes. This conjecture has been verified by computer for all positive even integers up to 1.6 ×1018. It is believed to be true by most mathematicians. • There are infinitely many primes of the form n2 + 1, where n is a positive integer. But it has been shown that there are infinitely many primes of the form n2 + 1, where n is either a positive integer or the product of at most two primes. • The Twin Prime Conjecture: The twin prime conjecture is that there are infinitely many pairs of twin primes. Twin primes are pairs of primes that differ by 2. Examples are 3 and 5, 5 and 7, 11 and 13, etc. The current world’s record for twin primes (as of mid 2011) consists of numbers 65,516,468,355∙2333,333 ±1, which have 100,355 decimal digits. INTEGERS, DIVISION, PRIMES

  34. Greatest Common Divisor • Definition: Let a and b be integers, not both zero. The largest integer d such that d | a and also d | b is called the greatest common divisor of a and b. It is denoted by gcd(a,b). • Example: gcd(24, 36) = ? • Example: gcd(17, 22) = ? • Example: gcd(10024, 0) = ? INTEGERS, DIVISION, PRIMES

  35. Greatest Common Divisor • Definition: The integers a and b are relatively prime if their greatest common divisor is 1. • Example: 17 and 22 • Definition: The integers a1, a2, …, an are pairwise relatively prime if gcd(ai, aj)= 1 whenever 1 ≤ i < j ≤ n. • Exercise: Which of the following are pairwise relatively prime? • 10, 17 and 21 • 10, 19 and 24 • 25, 26, 9 and 121 INTEGERS, DIVISION, PRIMES

  36. Finding gcd using prime factorizations • Suppose the prime factorizations of a and b are: where each exponent is a nonnegative integer, and where all primes occurring in either prime factorization are included in both. Then: • Example: • 120 = 23 ∙3 ∙5 500 = 22 ∙ 53 = 22 ∙ 30 ∙53 • gcd(120, 500) = 22 ∙30 ∙51 = 20 INTEGERS, DIVISION, PRIMES

  37. Euclidean Algorithm Euclid (325B.C. – 265B.C.) • The Euclidian algorithm is an efficient method for computing the greatest common divisor of two integers. • It is based on the fact that, if a > b, then gcd(a, b) = gcd(b, amodb). • Example: Find gcd(91, 287): • 287 mod 91 = 14, so gcd(91, 287) = gcd(91, 14) • 91 mod 14 = 7, so gcd(91, 14) = gcd(14, 7) • 14 mod 7 = 0, so gcd(14, 7) = gcd(7, 0) • gcd(7, 0) = 7 • Hence, gcd(91, 287) = 7 INTEGERS, DIVISION, PRIMES

  38. Least Common Multiple • Definition: The least common multiple of the positive integers a and b is the smallest positive integer that is divisible by both a and b. It is denoted by lcm(a,b). • The least common multiple can also be computed from the prime factorizations: If the prime factorizations of a and b are: where each exponent is a nonnegative integer, and where all primes occurring in either prime factorization are included in both. Then: • Theorem 5: Let a and b be positive integers. Then ab = gcd(a,b) ∙lcm(a,b) Proof: Exercise. INTEGERS, DIVISION, PRIMES

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