 
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 wholenumber multiples of n.
Solution: Suppose that nk + 1 is a multiple of n. Then for some q, nk + 1 = qn and so
1 = n(qk). Since n > 1 this is impossible and we see S = {nk+1 : k N} is an infinite
set of nonmultiples of n since each value of k corresponds to an additional example.
1.2 Let n be a whole number that is a wholenumber 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
