Math 3130, Abstract Algebra Homework 1 Key Summary: Math 3130, Abstract Algebra Homework 1 Key 1.1 Suppose that n is a positive whole number greater than 1. Prove that there are an infinite number of positive whole numbers that are not whole-number multiples of n. Solution: Suppose that nk + 1 is a multiple of n. Then for some q, nk + 1 = qn and so 1 = n(q-k). Since n > 1 this is impossible and we see S = {nk+1 : k N} is an infinite set of non-multiples of n since each value of k corresponds to an additional example. 1.2 Let n be a whole number that is a whole-number multiple of every other whole number. What values can n take on? Solution: zero. 1.3 The parity of a whole number is its evenness or oddness. The number two has even parity while the number three has odd parity. Prove that if x and y are whole numbers such that x + y has even parity then x and y have the same parity. Solution: Suppose that a + b = 2k (is even). Then if a = 2m is even as well and so b = 2(k = m) is also even. If, on the other hand, a is odd then a = 2m + 1 and so b = 2(k - m - 1) + 1 is also odd. 2 1.4 State the contrapositive of the fact proved in Problem 1.3. Solution: If two numbers do not have the same parity then their sum is not even. 1.13Prove that if two lines have the same Cartesian form they are the same line. Solution: Suppose that L, and M are both lines with Cartesian form y = mx + b. If Collections: Mathematics