10-26-2009Review for Exam IIYou should know how to apply all of the following problem solving strategies:1. Guess and check2. Tables3. Lists4. Patterns5. Algebra6. Work backwards7. Eliminate possibilities8. Pigeon hole principleYou should know all of the following definitions• Polya’s principle• Inductive reasoning• Deductive reasoning• Rule of direct reasoning• Rule of indirect reasoningYou should know all of the following number theoretic definitions• Divisibility• Factor• Multiple• Greatest common divisor• Least common multipleYou should know the divisibility tests for 2,3,4,5,7,8,10,11, and 13. You should be able to argue that ifa and b are divisible by c then a + b, a − b, and a · b are all divisible by c.Example Problems:1. Polya’s Principles(a) State Polya’s four principles.(b) With one or two sentences each, explain these four principles.2. Solve the following problem using the four Polya principles. You should indicate where and howyou are applying each step of the principles. Write up your answer as if you were presenting itto a classroom. That is, your write up should be brief (one or two sentences per step) and wellorganized.Problem. Find the nthterm of the sequence: 3, 5, 7, 9, 11, 13, ...3. Solve the following problem using the four Polya principles. You should indicate where and howyou are applying each step of the principles. Write up your answer as if you were presenting itto a classroom. That is, your write up should be brief (one or two sentences per step) and wellorganized.Problem. It takes Tom 4 hours to clean the garage and it takes Maria 3 hours. If they worktogether, how long will it take?4. Solve the following problem using the four Polya principles. You should indicate where and howyou are applying each step of the principles. Write up your answer as if you were presenting itto a classroom. That is, your write up should be brief (one or two sentences per step) and wellorganized.Problem. What is the sum of all the numbers in the 20th row of pascals triangle? How manynumbers are in this row? Solve the problem without writing down this row.5. (a) State the pigeonhole principle.(b) If there are 367 people in the education department, explain why at least two of them sharea birthday.(c) If there are 250,000 people in Lexington and any person can have at most 200,000 hairs ontheir head, explain why you can conclude that a least two people will have the same numberof hairs on their head.6. Mathematical Reasoning(a) Define inductive and deductive reasoning. Compare and contrast these two ideas.(b) We solved many types of problems using the pattern recognition strategy. What type ofmathematical reasoning do you use in this strategy? Explain.(c) Which type of reasoning does a detective typically use? Explain.(d) Suppose that you know a → b. If b is false what can you conclude and why?(e) Suppose that a → b is true and that b is true. Is a true? Explain and give an example.7. If a and b are whole numbers both divisible by c prove that a + b is also divisible by c. (Hint: First,write down what it means for a and b to be divisible by c)8. Show that the product of an even and an odd number must be an even number. (Hint: Use theuseful representations of even and odd numbers.)9. Is the number 36, 335, 936, 637 divisible by 7,13, or 11? Argue with the divisibility tests and showall of your work.10. (a) Give the definition for the greatest common divisor of the numbers a and b.(b) Give the definition for the least common multiple of the numbers a and c.(c) Find the greatest common divisor of the numbers 1092 and 525.(d) Find the least common multiple of the numbers 1092 and 525.11. Suppose that n = a · b and that the number c divides n. Is it always true that c divides a or b?Explain your answer.12. Consider the number abc, abc where the letters a, b, c represent digits. Argue that this number isdivisible by 7, 11, and 13.13. Mark the following as true or false. Explain each answer with one sentence, giving examples whereappropriate.(a) Every whole number can be written as a product of prime powers.(b) An even number times an even number is always odd.(c) If the statement “if p then q” is a true conditional statement and “q” is true then we mayconclude that “p” is true.(d) If the statement “if p then q” is a true conditional statement and “p” is true then we mayconclude that “p” is true.(e) One way to find the least common multiple of a and b is to just multiply a and b