187 – Quiz 2

Here is quiz 2.

Problem 1 asks to show that an integer is odd if and only if it is the sum of two consecutive integers.

Here is an argument: Let be an integer. We need to prove two statements, corresponding to the two directions of the “if and only if:”

1. If is odd, then there are two consecutive integers and such that
2. If is the sum of two consecutive integers and then is odd.

1. Assume that is odd. Then, by definition, there is an integer such that Note that so we can take and we see that and are consecutive, and

2. Now assume that where and are consecutive integers. Say, Then By definition, this means that is odd.

Problem 2 asks to show that if are integers and we have that and then also

To see this, assume that are integers such that and By definition, this means that there are integers and such that and Then But Let Then is an integer, and By definition, this means that

Problem 3 asks to show that if is a positive integer, then is composite.

As stated, this is false. For a counterexample, note that is a positive integer. However, is not composite according to the definition given in the book: 

An integer is composite if and only if there is an integer such that and

In any case, it is true that if is an integer and then is composite. To see this, note first that Now, if then and so Also, since then and therefore We have shown that and that By definition, this means that is composite.

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