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Homework 2 Math 295 (Fall 2016) Section 1.2 Problem 2 The differences between the squares of the positive integers 1, 4, 9, 16, 25, 36, 49 . . . are 3, 5, 7, 9, 11, 13, . . . . The differences between those squares which are two apart in the sequence are 8, 12, 16, 20, 24, . . . . The differences between those squares which are three apart are 15, 21, 27, 33, . . . . In other words, the difference between the nth square and the n+1st square is 1(2n+1); the difference between the nth square and the n + 2nd square is 2(2n + 2); and the difference between the nth square and the n + 3rd square is 3(2n + 3). In general, I hypothesize that the difference between the nth square and the n + k th square is k(2n + k). This can be proved fairly easily: for any n, k ∈ N, we have (n + k)2 − n2 = (n2 + 2nk + k 2 ) − n2 = 2nk + k 2 = k(2n + k). Exercise 3 The powers of 2 are 2, 4, 8, 16, 32, 64, 128, 256, . . . . The sequence of ones digits of these powers is 2, 4, 8, 6, 2, 4, 8, 6, . . . . 1 It looks like the ones digits will keep repeating in cycles of 2, 4, 8, 6. The powers of 3 are 3, 9, 27, 81, 243, 729, 2187, 6561, . . . . The sequence of ones digits of these powers is 3, 9, 7, 1, 3, 9, 7, 1, . . . . Here it looks like the ones digits will repeat in cycles of 3, 9, 7, 1. The similar thing about the two sequences of ones digits is that they both have a repeating period of 4; the difference is that different numbers are involved. If you try this for other one-digit numbers, you find that the ones digits of powers of 4 repeat in cycles of 4, 6; the ones digits of powers of 7 repeat in cycles of 7, 9, 3, 1; the ones digits of powers of 8 repeat in cycles of 8, 4, 2, 6; and the ones digits of powers of 9 repeat in cycles of 9, 1. (The ones digits of powers of 0,1,5, and 6 are always the same.) If we look instead at tens digits, for powers of 2 we get 0, 0, 0, 1, 3, 6, 2, 5, 1, 2, 4, 9, 9, 8, . . . and no clear pattern emerges with the limited sample. For the tens digits of powers of 3 we get 0, 0, 2, 8, 4, 2, 8, 6, 8, 4, 4, 4, 2, 6, . . . and again there is no clear pattern, although we only seem to obtain even digits. The tens digits of powers of 4 are 0, 1, 6, 5, 2, 9, 8, 3, 4, 7, 0, 1, 6, 5, . . . and here the 10-digit sequence 0, 1, 6, 5, 2, 9, 8, 3, 4, 7 seems to begin to repeat. Perhaps this has to do with the fact that the ones digits of powers of 4 repeat in cycles of only length 2? We’ll check the tens digits of powers of 9 to see: 0, 8, 2, 6, 4, 4, 6, 2, 8, 0, 0, 8, 2, 6, . . . . Here another 10-digit sequence, 0, 8, 2, 6, 4, 4, 6, 2, 8, 0, seems to begin to repeat. What about the tens digits of the powers of 5 and 6? The former are boring: 0, 2, 2, 2, 2, 2, 2, . . .. For the latter we get 0, 3, 1, 9, 7, 5, 3, 1, 9, 7, 5, 3, 1, 9, . . . which after the initial 0 seems to repeat in cycles of 3, 1, 9, 7, 5. Since the powers of 4 and 9 had units digits repeating in cycles of length 2 and their tens digits repeated in cycles of length 10, and the powers of 6 had units digits repeating in cycles of length 1 and their tens digits repeated in cycles of length 5, I hypothesize that the powers of 2,3,7, and 8 (which all had units digits repeating in 2 cycles of length 4) will have tens digits repeating in cycles of length 20. The tens digit of powers of 2 are easy to calculate, so we’ll find a few more of those: 0, 0, 0, 1, 3, 6, 2, 5, 1, 2, 4, 9, 9, 8, 6, 3, 7, 4, 8, 7, 5, 0, 0, 1, 3, 6, 2, 5 . . . . It does indeed seems as if the digits, after the initial zero, will repeat in cycles of length 20. Section 1.3 Exercise 2 (a) {1, 2, 3, 4, 5, 6, 7, 8, 9} (b) {2, 3, −3} (c) {{2, 4}, {2, 6}, {2, 8}, {2, 10}, {4, 6}, {4, 8}, {4, 10}, {6, 8}, {6, 10}, {8, 10}} Exercise 3 (a) i. Counterexample (since 2 is a prime and it is not odd). ii. Not a counterexample (since 6 is not prime). iii. Not a counterexample (since 9 is not prime and it is odd). (b) i. Not a counterexample (since 5 is not even). ii. Counterexample (since 2 is even and it is not a multiple of 4). iii. Not a counterexample (since 12 is a multiple of 4). (c) i. Not a counterexample (since {1, 2, 3} is not a subset of {5, 6, 7}). ii. Counterexample (since ∅ ⊂ ∅ and they have no element in common). iii. Not a counterexample (since {2} and Z have an element in common). (d) i. Counterexample (since (−3)2 = 9 and −3 ≤ 2). ii. Not a counterexample (since 3 > 2). iii. Not a counterexample (since 52 6= 9). (e) i. Not a counterexample (since ac > bd). ii. Not a counterexample (since a is not greater than b). iii. Counterexample (since a > b and c > d and ac ≤ bd). (f) i. Not a counterexample (since 0 is not odd). ii. Counterexample (since 3 is odd and 3 is not even). 3 Exercise 4 (a) If x is a real number, then x2 ≥ 0. (b) If (x, y) is in the first quadrant, then x > 0 and y > 0. (c) If S is a set with three elements, then no proper subset of S can have more than two distinct elements. (d) If l and m are lines in the plane which are not parallel, then l and m are identical or they meet in just one point. Exercise 5 (a) True. If an even number is a divisor of an odd number, then that even number is divisible by 2. Since 2 divides the even number and the even number divides the odd number, that would mean that 2 divides the odd number - i.e. that the odd number is even, a contradiction. (b) False. Counterexample: 6 is an even integer which has an odd divisor, 3. (c) False. Counterexample: 6 and 4 are even integers, but their quotient an integer. 6 4 is not even (d) False. Counterexample: Let a = −3, b = −2, and c = −2. Then a < b and a < c but a ≥ b + c. (e) True. Since a > b, and b > b − 1, we have a > b > b − 1, or a > b − 1. (f) False. Counterexample: Let a = 2, b = −3, and c = −4. Then a < bc since 2 < 12, but a ≥ b and a ≥ c. 4