# Let n be an integer Prove that n 2 leaves a remainder of 0,...

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21-127 Concepts of Mathematics Spring 2017 Homework 2 Due Thursday, February 2 Hand in complete solutions to all required problems (5 points each) at the beginning of recitation. You are not required to do or hand in bonus problems, though you should read and try them. You will receive a small bonus (one point each) for up to two complete and correct bonus problems. Papers typeset using L A T E X will also receive one bonus point. Required Problems 1. Let n be an integer. Prove that n 2 leaves a remainder of 0, 1 or 4 when divided by 5. 2. This problem concerns lattice points in 3-dimensional space. (a) Let k be the largest possible number of lattice points so that no pair of the points contains a lattice point on the segment between them. Find k and give an example of k such points. (b) Prove that in any set of k + 1 points that there must be some pair where the segment between them contains another lattice point. (c) Optional: Generalize this problem to n -dimensional space. 3. Let x be any real number such that x > -