hw1_1-1a_310-rem - Math 310 Section 2 Fall 2009 Remarks on...

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Math 310, Section 2, Fall 2009 Remarks on HW Section 1.1 From Section 1.1: 1a,b,c, 6, 8 1 Find the quotient and remainder when a is divided by b : (a) a = 302, b = 19 (b) a = - 302, b = 19 (c) a = 0, b = 19 Solution: (a) 302 = 19 · 15 + 17, q = 15, 0 r = 17 < b = 19. (b) - 302 = 19 · - 16 + 2, q = - 16, 0 r = 2 < b = 19. (c) 0 = 19 · 0 + 0, q = 0, 0 r = 0 < b = 19. 6 Use the Division Algorithm to prove that every odd integer is either of the form 4 k + 1 or of the form 4 k + 3 for some integer k . Proof: An even integer is, by definition, an integer divisible by 2, that is, an integer that has a remainder of 0 after division by 2. An odd integer is, then, an integer not divisible by 2, that is, an integer that has a remainder of 1 after division by 2. Let a be an odd integer. Apply the Division Algorithm to a and b = 4: there exist unique integers q and r such that a = 4 q + r where 0 r < 4. That is, r is one of 0, 1, 2, or 3. We will consider each of these four cases and divide a by 2 to see what the remainder is: If r = 0, then a = 4 q = 2(2 q ) and so the remainder is 0 and a is even.
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