Fall 2017 Test II review problems Dr. Holmes October 18, 2017 This is a quite miscellaneous grab bag of relevant problems from old tests. Some are certainly repeated. 1. Give the complete addition and multiplication tables for mod 4 arithmetic. + 0 1 2 3 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2 0 1 2 3 0 0 0 0 0 1 0 1 2 3 2 0 2 0 2 3 0 3 2 1 Give the complete addition and multiplication tables for base 4. + 0 1 2 3 0 0 1 2 3 1 1 2 3 10 2 2 3 10 11 3 3 10 11 12 1
0 1 2 3 0 0 0 0 0 1 0 1 2 3 2 0 2 10 12 3 0 3 12 21 Notice that these are not the same thing! Convert 27 and 45 to base 4. Carry out the calculations 27+45 and (27)(45) in base 4. Convert your results back to base 10 and check that you have the right answers. 27 = 16+8+3 = 123 4 45 = 32+12+1 = 231 4 adding: 1 1 1 2 3 2 3 1 1 0 2 0 and you can check that 24+45=72 = 1020 4. 1 2 3 2 3 1 1 2 3 1 1 2 0 2 1 3 1 1 1 2 1 0 2 3 3 3 and 102333 4 = 1215 ten = (27 ten ) (45 ten ). Fall 2017 comment: We will not be doing actual calculations in bases, just conversions. 2
2. Compute gcd(6105, 21390). 21390 divided by 6105 is 3 remainder 3075. 6105 divided by 3075 is 1, remainder 3030. 3075 divided by 3030 is 1, remainder 45. 3030 divided by 45 is 67, remainder 15. 45 divided by 15 is 3 remainder 0. So the gcd is 15. Determine integers m and n such that 6105m+21390n = gcd(6105, 21390). 21390 - (3)(6105) = 3075. 3030 = 6105-3075 = 6105 - (21390 - (3)(6105)) = (4)(6105) - 21390 45 = 3075-3030 = (21390 - (3)(6105)) - ((4)(6105) - 21390) = (2)(21390) - (7)(6105) 15 = 3030 - (67)(45) = ((4)(6105) - 21390) - (67)((2)(21390) - (7)(6105)) = (4 + (67(7))(6105) - (1 + (67(2)))(21390) = (473)(6105) - (135)(21390) So gcd(21390,6105) = 15 = 473*6105-135*21390. Fall 2017 comment: Put this in table format. 3
3. Do one of the two induction proofs. (a) Prove by mathematical induction that the sum n 3 n = 1 + 3 + 9 +... + 3 n i=0 is equal to 3n+1 1 2, for each natural number n 0. Be sure to clearly identify the basis step and the induction step. In the induction step, identify the inductive hypothesis and the goal to be proved, and indicate where in your proof the inductive hypothesis is used. A significant amount of credit will be based on setting up the proof correctly. 4
(b) Prove by mathematical induction that 10 n 1 is divisible by 3 for each natural number n 0. Be sure to clearly identify the basis step and the induction step. In the induction step, identify the inductive hypothesis and the goal to be proved, and indicate where in your proof the inductive hypothesis is used. A significant amount of credit will be based on setting up the proof correctly. 5
4. Do two of the three following proofs (two are induction, one is proof by contradiction (omitted). In the math induction proofs, clearly identify the basis step, induction step, and induction hypothesis, and show me where the induction hypothesis is used in your proof. If you do all parts your best two will count and extra credit is possible. (a) Prove by mathematical induction that the sum of the first n odd numbers is equal to n 2. This sum can be written in the form ni=1 (2i 1) or the form 1 + 3 + 5 +... + (2n 1) 6
(b) Prove by mathematical induction that for any natural number n, n 3 + 5n is divisible by 3. 7
5. The sequence G n is defined by the recurrence relation G 1 = 2; G 2 = 3; G n = 2G n 1 + 8G n 2 for n > 1 Compute the next four terms of the sequence using the recurrence relation. A closed form expression for G n, believe it or not, is ( 7 24 )(4n ) 5 12 ( 2)n. Check that the first few values are correct. If you feel up to the algebra, try proving by induction that the formula is correct (the algebra is a bit too unpleasant for a test). 8
6. Compute the greatest common divisor of 2701 and 629. Show all steps. Clearly identify the number which you claim is the gcd! Use your gcd calculation to find integers x and y such that 2701x + 629y = gcd(2701,629) (and include a check that they work). 9
7. Prove by mathematical induction that the sum of the first n odd numbers is n 2 : 1 + 3 + 5 +... + (2n 3) + (2n 1) = n 2. As part of your setup, write this statement in summation (Σ) notation: you do not have to use this notation in your proof, though. Be sure that the basis step, induction step, inductive hypothesis and any use of the inductive hypothesis in the proof are clearly identified. Make sure that you clearly distinguish between what you are assuming and what you are trying to prove. 10
8. Prove by mathematical induction that n 3 + 5n is divisible by 3 for each natural number n. Be sure that the basis step, induction step, inductive hypothesis and any use of the inductive hypothesis in the proof are clearly identified. Make sure that you clearly distinguish between what you are assuming and what you are trying to prove. 11
9. Compute the first six terms of the sequence defined by the equations a 1 = 1; a 2 = 3; a n+2 = 6a n+1 8a n. A formula for a n is 1 8 (4n ) + 1 4 (2n ). Check that this works for the first few terms. Use the formula to compute a n. Prove that the formula is correct by induction. 12
10. Compute 38 div 10 and 38 mod 10. (div is integer division and mod is remainder.) That was easy! Compute -38 div 10 and -38 mod 10. obvious. That should be slightly less 13
11. Present the multiplication table for mod 5 arithmetic. 14
12. (a) Compute gcd(11, 37). Express gcd(11,37) in the form 37x + 11y. (b) Compute the gcd of 30 and 74 using the Euclidean algorithm (and clearly indicate that you know what the answer is: gcd(30, 74) =...). Find integers x and y such that 30x + 74y = gcd(30, 74). 15
13. Prove by mathematical induction that for any nonnegative integer n, the sum Σ n i=03 i = 1 + 3 + 9 +... + 3 i 1 + 3 i = 3n+1 1 2. 16
14. Use the Euclidean algorithm to find the greatest common denominator of 23 and 37, showing all work and clearly indicating to me that you know what the answer is: gcd(23, 37) =.... Determine x and y such that 23x + 37y = 1: this should come out of your gcd calculation. Show all work. 17
15. Prove by mathematical induction that the sum 2 + 4 + 6 +... + 2n of the first n even numbers is n(n + 1). Be sure to clearly identify the basis step, induction hypothesis and induction goal and clearly label the place(s) where the induction hypothesis is used. 18
16. The Fibonacci numbers are defined by the following conditions: F 0 = 1; F 1 = 1; F n+2 = F n+1 + F n. Prove the following statement by mathematical induction: for each natural number n, F 3n is odd, F 3n+1 is odd, and F 3n+2 is even (notice that this completely describes the pattern determining what Fibonacci numbers are even and what Fibonacci numbers are odd). Hint: this is an ordinary induction (you need just one basis step and just one induction hypothesis). If you write out correctly exactly what you assume and exactly what you need to show, it is not hard. Write English sentences! 19
17. Compute the first six terms of the sequence defined by the equations 18. A sequence is defined by a 0 = 1; a 1 = 3; a n+2 = 6a n+1 8a n. Compute its first five terms. a 0 = 3; a 1 = 1; a n+2 = 2a n+1 + 3a n 19. A sequence a n is defined by the recursive definition a 1 = 3; a 2 = 5; a k+2 = 2a k+1 a k Compute the first ten terms of this sequence. You should recognize this sequence! Give a closed form formula for a n. Prove this to be correct by math induction. 20
20. Prove by mathematical induction that the sum of the first n integers, is equal to n i = 1 + 2 + 3 +... + n i=1 n(n + 1) 2 21. Compute the first six terms of the sequence defined by a 0 = 2; a 1 = 3; a n+2 = 3a n+1 2a n. 22. Determine the gcd of 123 and 111 and numbers x and y such that 123x + 111y = gcd(123, 111) 23. Compute the first six terms of the sequence defined by a 0 = 4; a 1 = 7; a n+2 = 3a n+1 2a n. 21
24. Write out the multiplication table for mod 7 arithmetic. 25. Compute gcd(38, 71) using the Euclidean algorithm. Express gcd(38, 71) in the form 38x + 71y where x and y are integers. 22
26. Do two of the three mathematical induction problems. These count as two separate problems on the test, not as two parts of one question. If you do all three, your best two problems will count, and success on all three may yield some extra credit (don t try for a third until you have completed the rest of the test!) (a) Prove by mathematical induction that the sum of the first n numbers is n(n+1) 2. 23
(b) Prove by mathematical induction that for each natural number n, 4 n 1 is divisible by 3. You may use the fact that the sum or difference of two numbers both divisible by 3 is also divisible by 3. 24
(c) Prove by mathematical induction that the sequence defined by a 0 = 3; a 1 = 7; a n+2 = 5a n+1 6a n satisfies a n = 2 n+1 + 3 n for each natural number n. 25
27. Do both of the math induction problems. The one on which you do better will count twice as much as the other. In each problem, clearly label the basis step, the induction hypothesis, and the goal of the induction step. Indicate clearly where the induction hypothesis is used in the proof of the induction step. (a) Prove by induction that the sum of the first n odd numbers is n 2, for any positive integer n. 26
(b) Prove by induction that 10 n 1 is divisible by 9 for each natural number n. 27
28. Compute the first six terms of the sequence with the recursive definition a 0 = 1; a 1 = 2; a n+2 = 4a n+1 3a n. 29. Use the Euclidean algorithm to compute gcd(180, 125); further, use the Euclidean algorithm computation to find integers x and y such that 180x + 125y = gcd(180, 125). Use the table format used in class examples and show all work. Your answer must clearly show that you know what gcd(180, 125) is, what x is, and what y is. 28
30. Prove by mathematical induction that n 3 + 2n is divisible by 3 for each natural number n. The basis step, induction step and the place where you use the induction hypothesis should all be clearly identified. 29
31. A sequence a n is defined by the recursive definition a 1 = 3; a 2 = 5; a k+2 = 2a k+1 a k Compute the first ten terms of this sequence. You should recognize this sequence! Give a closed form formula for a n. For extra credit, or to replace one of the two previous problems, prove this to be correct by math induction. 30
32. Multiplication table for mod 5 arithmetic: 0 1 2 3 4 0 0 0 0 0 0 1 0 1 2 3 4 2 0 2 4 1 3 3 0 3 1 4 2 4 0 4 3 2 1 Find a solution to 137x + 7y = 1 in integers. Here s the Euclidean algorithm calculation: 7x + 137y x y q 137 0 1 7 1 0 4 19 1 19 3 20 1 1 1 39 2 1 31