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WUCT121 Discrete Mathematics Numbers Tutorial Exercises 1. Natural Numbers 2. Integers and Real Numbers 3. The Principle of Mathematical Induction 4. Elementary Number Theory 5. Congruence Arithmetic WUCT121 Numbers Tutorial Exercises 1 Section 1: Natural Numbers Question1 What is a closed operation? Question2 What are the closed operations on Õ? Question3 What operations have an identity in Õ and what are they? Question4 For those operations which have an identity, which elements of Õ have inverses and what are they? Question5 For the following, state which property is being demonstrated: (a) a + b = b + a , a , b ∈ . (b) 4( x + y ) = 4 x + 4 y, x, y ∈ . (c) 5 × (4 × 3) = (5 × 4) × 3 = 20 × 3 = 60 . Question6 Simplify the following, indicating which property you are using at each step: 8( x + y ) + ( y + x )15 . Question7 State the Law of Trichotomy. Question8 Use the definition of “<” to show 12 < 25. Question9 What is the property of transitivity? Question10 Show if a < b, then ac < bc where a , b, c ∈ . Question11 State the well-ordering property for Õ. WUCT121 Numbers Tutorial Exercises 2 Section 2: Integers and Real Numbers Question1 What are the closed operations on ? Question2 What operations have an identity in and what are they? Question3 For those operations which have an identity, which elements of have inverses? Question4 Is well-ordered? Demonstrate why or why not Question5 Use the definition of odd to show if –67 is odd Question6 What is a prime number? Question7 What is a composite number? Question8 What is a rational number? Question9 What are the closed operations on ? Question10 Which operations on have identities and what are they? Question11 For those operations which have an identity, which elements of have inverses? Question12 WUCT121 Is well-ordered? Demonstrate why or why not. Numbers Tutorial Exercises 3 Section 3: The Principle of Mathematical Induction Use Mathematical Induction to prove the following: Question1 For all n ∈ , ( n + 1)!≥ 2n n −1 Question2 For all n ∈ , cos x × cos 2 x × cos 4 x × L × cos 2 Question3 ∀n ∈ ,12 + 2 2 + L + n 2 = Question4 ∀n ∈ , ∃z ∈ : 4 n − 1 = 3z Question5 ∀n ∈ , n 3 ≥ 5n − 4 Question6 Prove that for every real number x and each n ∈ , (1 + x ) n ≥ 1 + nx . Question7 Prove that 2 + 6 + 10 + K + ( 4n − 2) = 2n 2 for all n ∈ . Question8 Prove that 7 n − 2 n is divisible by 5 for all n ∈ . Question9 Find the mistake in the following proof fragment. Claim(l) is 12 = 2 n sin x n(n + 1)(2n + 1) 6 Let Claim(n) be: 12 + 2 2 + L + n 2 = Step 1: x= sin 2 n x n(n + 1)(2n + 1) for all n ∈ . 6 1(1 + 1)(2 × 1 + 1) 6 LHS = 1, RHS = 1, so LHS = RHS. Therefore, Claim(1) is true. Step 2: Assume that Claim(k) is true for some k ∈ k (k + 1)(2k + 1) that is, k 2 = 6 (1) Prove Claim(k + 1) is true; that is, prove that (k + 1)2 = WUCT121 (k + 1)((k + 1) + 1)(2(k + 1) + 1) Numbers Tutorial Exercises 6 4 Question10 Let Claim(n) be: For all integers n ≥ 1, 3n − 2 is even. What is wrong with the following proof ? Suppose the Claim(k) is true for some k ∈ . That is, suppose that 3k − 2 is even. We must show that Claim(k + 1) is true. That is, we must show that 3k +1 − 2 is even. Now, 3k +1 − 2 = 3 × 3k − 2 = (1 + 2) × 3k − 2 = 2 × 3k + 3k − 2 Now 3k − 2 is even by the hypothesis and 2 × 3k is even by inspection. Hence, the sum of the two quantities is even. Therefore, Claim(k + 1) is true. Therefore, Claim(n) is true for all n ∈ . Question11 Let Claim(n) be the statement “ n 2 + n + 11 is a prime number”. (a) Verify that Claim(n) is true for n = 1, 2, 3, 4, 5, 6, 7, 8, and 9. (b) Can we conclude that Claim(n) is true for all n ∈ ? Explain. Question12 Try to use induction to prove that 1 + (a) 1 1 1 + +K+ ≤ 2 for all n ∈ . 2 4 2n What goes wrong? Prove that 1 + (b) 1 1 1 1 + +K+ ≤2− for all n ∈ . n 2 4 2 2n thus, showing that 1 + 1 1 1 + +K+ ≤ 2 for all n ∈ . 2 4 2n Question13 1⎞ ⎛ ⎛ 1 ⎞⎛ Prove that ⎜1 + ⎟⎜1 + ⎟ K ⎜1 + 2⎠ ⎝ ⎝ 1 ⎠⎝ Question14 Prove ∑ 2 i = 2(2 n − 1) 1⎞ ⎟ = n + 1 for all n ∈ . n⎠ n WUCT121 i =1 Numbers Tutorial Exercises 5 n 1 n = i =1( 4i − 3)(4i + 1) 4n + 1 Question15 Prove ∑ Question16 Prove n3 − 4n + 6 is divisible by 3 for all n ∈ Question17 Use the identity ∑ i = Question18 Prove ∑ (2i − 1)2 = Question19 Prove ∑ i (i + 2) = Question20 Prove 1 + 3 + K + (2n − 1) = n 2 for all n ∈ . Question21 If bn = 3bn −1 , n ≥ 2, b1 = 3 . Prove bn = 3n for all n ∈ . Question22 If un = un −1 + n − 1, n ≥ 2, u1 = 0. Prove un = Question23 If un = 2un −1 + 1, n ≥ 2, u1 = 1. Prove un = 2 n − 1 ∀n ∈ . Question24 If sn = ( −n) sn −1 , n ≥ 2, s1 = −1 . Prove sn = ( −1) n n! , ∀n ∈ . Question25 If un = 5un −1 − 6un − 2 , n ≥ 3, u1 = 2, u2 = 4 . Prove un = 2 n , ∀n ∈ . Question26 If un = 5un −1 − 6un − 2 , n ≥ 3, u1 = 1, u2 = 5 . Prove un = 3n − 2 n , ∀n ∈ . n i =1 n i =1 n i =1 2 n ⎛ n ⎞ n( n + 1) to prove that ∑ i 3 = ⎜⎜ ∑ i ⎟⎟ 2 i =1 ⎝ i =1 ⎠ n(2n − 1)(2n + 1) 3 n( n + 1)(2n + 7) 6 n( n − 1) , ∀n ∈ . 2 Question27 Let u1 , u2 ,..., un ... be real numbers such that un = −2un −1 + 3un − 2 , n ≥ 3, u1 = 2, u2 = 2 . Prove that un = 2 for all n ∈ . Question28 WUCT121 If an = 2an −1 − an − 2 , n ≥ 3, a1 = 3, a2 = 5. Prove an = n + 2, ∀n ∈ . . Numbers Tutorial Exercises 6 Section 4: Elementary Number Theory Question1 (a) Let m, n ∈ . Is 6m + 8n even? Why? (b) Let r , s ∈ . Is 6r + 4s 2 + 3 odd? Why? (c) Let u, v ∈ , with u > v. Is u 2 − v 2 composite? Why? Question2 Write down the definition of divisibility. Question3 (a) Is 52 divisible by 13? Why? (b) Let k ∈ . Is 3 a divisor of (3k + 1)(3k + 2)(3k + 3) ? Why? (c) Does 7 | 13 ? Why? (d) Let a, b ∈ . Does 4 | (6a × 10b) ? Why? Question4 State the Quotient Remainder Theorem. Question5 State the Fundamental Theorem of Arithmetic. Question6 Use a sieve of Eratosthenes to find the primes between 201 and 300 Question7 List the twin primes between 201 and 300. Question8 Find the prime factorization, gcd and lcm for the following pairs of numbers: (a) 369, 8619 (c) 1375, 605 (b) 936, 7623 (d) 4968, 9000 Use the Euclidean Algorithm to find the gcd(a, b) for the following pairs of numbers. Find values m, n ∈ . such that gcd(a, b) = am + bn Question9 (a) a = 72, b = 63. (e) a = 63, b = 24. (b) a = 2104, b = 21. (f) a = 336, b = 60. (c) a = 15, b = 10. (g) a = 7684, b = 4148. (d) a = 9, b = 5. (h) a = 90, b = –54. WUCT121 Numbers Tutorial Exercises 7 Question10 Give three pairs of numbers that are relatively prime. Question11 Fermat Primes are given by the formula F (n) = 22 + 1, for n ∈ . n (a) Write down the first three Fermat Primes. Are they all prime? (b) Find F(4) and F(5). Are they prime? Question12 Mersenne Primes are given by the formula g (n) = 2n − 1, for n ∈ . (a) Find g(n) for 2, 3, 4, 5, 6, 7, 8, 9. (b) Which of the values for g(n) found in part (a) are prime? (c) For what values of n do you suggest that g(n) will give a prime number? (d) Find g(11). How does this compare with your opinion for part (c)? (e) Prove that whenever n is composite, then g(n) is also composite. [Hint: Let n = pq, for some p, q ∈ , and factorise g(n).] Question13 What can you say about 3n − 1, for n ∈ ? Will this produce prime numbers? Why or why not? Question14 Show that in any group of 367 people at least two people must have the same birthday. Does it work for 366 people? Question15 Seven different integers are chosen from 1 to 12 inclusive. Must at least one be odd? Question16 Let T = {1, 2, 3, 4, 5, 6, 7, 8, 9} . Five integers are chosen from T. Must there be two which add to 10? Question17 A programmer writes 500 lines of computer code in 17 days. Must there be at least one day the programmer wrote 30 or more lines of code? Question18 Show that at a party of 10 people at least two of the people must have the same number of friends at the party. You may assume friendship is commutative, that is A is friend of B ⇔ B is friend of A. WUCT121 Numbers Tutorial Exercises 8 Section 5: Congruence Arithmetic Question1 (a) Find x ∈ , 0 ≤ x < 11 so that 129 ≡ x(mod11) . (b) Find x ∈ , 0 ≤ x < 13 so that 311 ≡ x(mod13) . (c) Find x ∈ , − 11 < x ≤ 0 so that 4 ≡ x(mod11) . Question2 Use the definition of congruence modulo to prove the following properties for congruence modulo 3. (a) ∀a ∈ , a ≡ a (mod 3) . (Reflexive Property) (b) ∀a, b ∈ , a ≡ b(mod 3) ⇒ b ≡ a(mod 3) (Symmetric Property) (c) ∀a, b, c ∈ , a ≡ b(mod 3) ∧ b ≡ c(mod 3) ⇒ a ≡ c(mod 3) Question3 (Transitive Property) Let n = 7 . (a) For m ∈ , is [m] (in 7 ) a number, a set of numbers or a set of sets? (b) Describe [2], [5] and [7] in 7 by listing eight elements in each set. Write down the set 7 . (c) True or false (in 7 ) ? (i) [2]=[9] (iv) 3 ∈7 (ii) [5] ≡ 5(mod 7 ) (v) [10] ∈7 (iii) 5 ∈ [19] Question4 (vi) a ≡ b(mod 7 ) ⇒ [ a ] ∈ [b] Write out the addition and multiplication tables for 5 , 6 and 1 . (a) Do all elements in 5 have multiplicative inverses? (b) Do all elements in 6 have multiplicative inverses? (c) Let [m] ∈6 have an inverse under multiplication, What can you say about gcd(m, 6) Question5 WUCT121 Let n ∈ . Prove that [a ] = [b] ⇔ a ≡ b(mod n ) Numbers Tutorial Exercises 9 Question6 In 11 , (a) Find values of m and n such that 410 ∈ [m] and 411 ∈ [n] . Hint: Find m ∈ , 0 ≤ m < 11 so that 410 ≡ m(mod11) (b) Find values of i and j such that 310 ∈ [i ] and 311 ∈ [ j ] . (c) Based on your results from part (a) and (b), can you make a conjecture about x10 (mod11) and x11(mod11) , where x ∈ and 0 < x < 11 ? Question7 Let n = 9 . (a) Write out the addition and multiplication tables for 9 . (b) True or false (in 9 ) ? (i) [3]=[–13] (v) The element [1] is the identity of 9 under addition (ii) − 9 ∈ [19] (vi) Every element of 9 has a (iii) 2 ∈9 (iv) 4 ≡ 9(mod 5) ⇒ 4 ∈ [5] (in 9 ) Question8 multiplicative inverse. Every natural number m can be written in the form m = d k × 10k + d k −1 × 10k −1 + K + d 2 × 100 + d1 × 10 + d 0 , where di ∈ . is the digit in the 10ith position of m. 7526 = 7 × 1000 + 5 × 100 + 2 × 10 + 6 Examples : 57402 = 5 × 10000 + 7 × 1000 + 4 × 100 + 0 × 10 + 2 (a) Let S be the sum of the digits of 7526. Find S. Find x in each case, where 0 ≤ x < 9 (i) 7 × 1000 ≡ x(mod 9 ) (ii) 5 × 100 ≡ x(mod 9 ) (iii) 2 × 10 ≡ x(mod 9 ) (iv) (7 × 1000 + 5 × 100 + 2 × 10 + 6) ≡ x (mod 9 ) Deduce that 7526 ≡ S (mod 9 ) . WUCT121 Numbers Tutorial Exercises 10 (b) Let S be the sum of digits of 57402. Show that 57402 ≡ S (mod 3) . (c) Prove that for any integer m, if the sum of the digits of m is S, then m ≡ S (mod 3) and m ≡ S (mod 9 ) . Hint: Use the fact that 10n ≡ 1(mod 3) and 10n ≡ 1(mod 9) (d) What divisibility rules have just been shown? WUCT121 Numbers Tutorial Exercises 11