Assignments for Math 2030 then the hard copy will not be correct whenever your instructor modifies the assignments. exams, but working through the problems is a good way to prepare for the exams. It is also important to know the examples that your instructor covers in class. Section Problems Exponents Simplify the expressions as much as possible leaving the answer free of negative exponents. Find the following greatest common divisors and least common multiples by obtaining the prime factorizations of each number. 6. lcm(504,2079), 7. gcd(504,2079) 8. lcm(301,1849), 9. gcd(301,1849) 10. lcm(1505,18533), 11. gcd(1505,18533) Find an exact simple fraction for each of the following. A simple fraction is in the form where p and q are integers. Write each of the following numbers as sums of powers of 2 using the smallest number of powers of 2 that is possible. 14. 1059, 15. 164351 16. Bob needs to get a job real bad since otherwise his wife is going to change the locks to their apartment. Bob tells a potential employer that he is willing to just accept pennies if he will give him a job. His employer agrees to pay him one penny for the first day, 2 pennies for the second day, 4 pennies for the third day, 8 pennies for the fourth day, and so on. Since his employer usually pays his employees about $40 a day his employer agrees to this plan. For how many working days will Bob keep this job? How many days until Bob is making $50000 a day if his employer likes Bob a lot? 17. Find a counterexample to the statement where a and b are positive integers.
18. Find a counterexample to the statement where a is an integer. Combinations and Binomial Theorem 6. Compute the integer coefficient of if we expand. 7. Compute the integer coefficient of if we expand. 3 Difference Quotients Compute the difference quotient for each of the following functions. Use the formula given below and do the algebra correctly showing all steps down to the step where the h in the denominator is cancelled. Find one counterexample for each of the following statements. Your counterexample can be any integers for and, but for you must use a nonzero integer. 1. 2. 3. 4 1.1 Set Theory 5 3.1 Hashing Functions 6 1.2 Propositions Find the DNF for for pnr if we have the three vaiables p, q, and r.
Write the DNF form for the following table. (?) T T T T T T F F T F T F T F F T F T T F F T F F F F T T F F F T 7 1.3 Conditional Propositions Translate each of the following expressions to an expression that contains only nand. Find a specific counterexample to the assertion that equivalent to. is 1.5 Quantifiers 9 1.6 Nested Quantifiers 3.2 Sequences 11 Preview of Mathematical induction 1. Each of the following questions concerns the predicate Each of the answers must have the property that it is true or false, but don't just write true or false. a. What is P(1)? a. What is P(5)? a. What is P(10)? a. What is P(n+2)? 2. Each of the following questions concerns the predicate
Each of the answers must have the property that it is true or false, but don't just write true or false. a. What is P(1)? a. What is P(5)? a. What is P(10)? a. What is P(n+2)? 3. Each of the following questions concerns the predicate for every integer n. Each of the answers must have the property that it is true or false, but don't just write true or false. a. What is P(1)? a. What is P(5)? a. What is P(10)? a. What is P(n+2)? 2.4 Mathematical Induction 13 5.1 Divisors What are the quotient and remainder when a. 47 is divided by 8 b. -2002 is divided by 87 c. -2 is divided by 7 d. 0 is divided by 10 Evaluate these quantities(no negative answers) a. -17 mod 3 b. 207 mod 15 c. -2003 mod 3 d. 10277 mod 12 Find 5 different integers that are congruent to 4 modulo 12. What does a 12-hour clock read a. 100 hours after it reads 2:00 b. 5000 hours after it reads 12:00 What does a 24-hour clock read a. 100 hours after it reads 13:00 b. 5000 hours after it reads 23:00 c. 5000 hours after it reads 10:50 d. 700 days after it reads 10:50 ab then p a or p b. Assume that p, a, and b are positive integers greater than 1. What extra condition on p would guarantee that the above assertion is true.
2.4 Mathematical Induction Use mathematical induction to prove each of the following facts: a. Show that 3 divides where n is a nonnegative integer. b. Show that 5 divides where n is a nonnegative integer. Find specific counterexamples to the following statements: a. All prime numbers are odd numbers. b. The perimeter of a rectangle is never an odd number. c. If n is an integer and is divisible by 4 then n is divisible by 4. 15 6.1 Basic Principles of Counting 1, 2, 3, 4, 7, 13, 14, 16, 20, 23, 26, 28, 30, 31, 34, 37, 38, 41,44, 45, 46, 47, 50, 52, 55, 58, 64, 67, 74 6.2 Permutations and Combinations 1-4, 7, 10, 13, 19, 21, 24, 25, 28, 31, 33, 36, 39, 43, 44, 45, 46, 48, 49 3.3 Relations 3.4 Equivalence Relations 5.2 Integer Algorithms Compute the following quantities carefully. (Any calculator errors are your fault.) a. b. c. d. 1. Using addition of ' mod 26 to find the ciphertext corresponding to each of the following messages. In each case the ciphertext should be a sequence of numbers separated by commas. (a) Let and the message is "Jane is guilty". (b) Let and the message is "Use the fifth drop". 2. Using multiplication by ' mod 26 to find the ciphertext corresponding to each of the following messages. In each case the ciphertext should be a sequence of numbers separated by commas. (a) Let and the message is "Jane is guilty". (b) Let and the message is "Use the fifth drop".
3. Using subtraction of ' mod 26 to find the plaintext corresponding to each of the following secret messages. In each case the plaintext should be words that might appear in some English dictionaries. (a) Let and the message is "23 8,5,1 25,23,10 10,11,16 8, 5,18,1". (b) Let and the message is "6,20,17,11 25,13,23,17 6,20,17 20,21,19,20,5 20,21,19,20,17,4 13,0,16 6,20,17 23,1,9,5 25,1, 4,17 18,4,17,3,7,17,0,6". 4. Using multiplication by ' mod 26 to find the plaintext corresponding to each of the following secret messages. In each case the plaintext should be words that might appear in some English dictionaries. (a) Let and the message is "0 11,1,2,14,13,17 5,0,24 21,5, 14 24,0,16,14 14,13,14,16,2,14,24". (b) Let and the message is "20,19 8,22,24 18,0,13 13,22,15 10,25,5,21,0,20,13 8,22,24 1,22 13,22,15 24,13,1,10,23,6,15,0, 13". 5.3 Euclidean Algorithm Find the gcd for problems 1,4,7,10 by factoring the numbers. Find the gcd for problems 1,4,7,10 by using the Euclidean Algorithm. For each of the above pairs of integers find the integers s and t so that For each of the above pairs of integers find the lcm. Use the Euclidean Algorithm to do each of the following computations. (a) Find the inverse of 17 modulo 26. (b) Find the inverse of 3 modulo 26. (c) Find the inverse of 2 modulo 26. (wierd problem) (d) Find the inverse of 81 modulo 98. (d) Find the inverse of 8 modulo 13. 3.2 Strings In the following exercises we are looking at strings in. ALL derivations should begin with S, but derived strings must not contain an S. derived? derived?