# Solutions to Problem Set 12

Math 310-01/02

10-6-2017

1. Prove that if n is an integer, then is not divisible by 3.

If n is an integer, then by the Division Algorithm one of the following 3 cases holds:

If , then

Hence, is not divisible by 3.

If , then

Hence, is not divisible by 3.

If , then

Hence, is not divisible by 3.

Therefore, if n is an integer, then is not divisible by 3.

2. Prove that if x is a real number, then

Consider the cases , , and .

Case 1. .

In this case, , so . Also, certainly implies , so , and . Therefore,

Thus, in this case, is true.

Case 2. .

In this case, , so . Also, , so , and . Therefore,

Now gives , so .

Thus, in this case, is true.

Case 3. .

In this case, , so . Also, certainly implies , so , and . Therefore,

Thus, in this case, is true.

Since the inequality holds in all cases, it is true.

I don't know that I have ever found any satisfactory answers of my own. But every time I ask it, the question is refined. ... questioning as exploration, rather than the search for certainty. - Ta-Nehisi Coates

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