Dr. Cordero
Collaboration on this test is both allowed and encouraged. Each student should write up his or her own solutions to the problems on this test and turn them in.
You will be graded both on mathematical content and on clarity of expression. Some of the questions are a bit open-ended so be creative, make conjectures and back up your assertions with a proof or a counterexample. In writing your answers use complete English sentences and be sure to say exactly what you mean. Papers will be graded on the basis of what you have written, so be sure to take the time to express yourself clearly. If you are stuck on a problem and have no idea where to begin, a good way to get started is to look at lots of numerical examples and try to find a pattern.
Please use standard 81/2 by 11 white paper (lined or unlined). Start each problem at the top of a fresh page (not on the back of a page you have already written on) and write your name on every page. Staple the whole test together.
- Find the q and r guaranteed by the Division Algorithm.
- a=36, b=67
- a=-15, b=4
- a=48, b=-15
- The set of all odd positive integers can be divided into two classes, depending upon their remainders when divided by 4:
- All those which are of the form 4n+1, namely 1, 5, 9, 13, etc. let’s call these the four-one integers.
- All those which are of the form 4n+3, namely 3, 7, 11, 15, etc. Let’s call these the four-three integers.
In class we proved that the product of two odd numbers is odd, the product of two even numbers is divisible by 4, and the product of an odd and an even is even. What kinds of analogous statements can one make about the product of two four-one numbers, two four-three numbers, or a four-one and a four-three? (You should be able to say more that just that they are even or odd. Don’t forget proofs!)
- There are many examples of triples of positive integers a, b, and c for which
. For example, 
or 
. But none of the commonly known examples have both a and b odd. Why?
Either find an example with both a and b odd, or show why
is never a perfect square when both a and b are odd. (Hint: Look at the notes on odd and even numbers and consider remainders when you divide by 4.) - For the first few odd integers
is a multiple of 24. (Verify it!)
Is always a multiple of 24 when n is an odd integer? Explain why or give an example to show that this not always true. - An equation of the form ax+by=c is the equation of a straight line in the plane. A point in the plane with both coordinates integers is called a lattice point. The x and y axes divide the plane into quadrants, with the first quadrant being where both x and y are positive.
- Draw a careful graph of the line 3x+4y=7 in the plane. Does the graph pass through any lattice points in the first quadrant? If so, how many? Justify your answer.
- Draw a careful graph of the line 7x+10y=6 in the plane. Does the graph pass through any lattice points in the first quadrant? If so, how many? Justify your answer.
- Draw a careful graph of the line 3x-7y=10 in the plane. Does the graph pass through any lattice points in the first quadrant? If so, how many? Justify your answer.
- Formulate a conjecture, in terms of a, b and c for when the graph of the line ax+by=c will pass through infinitely many lattice points in the first quadrant. You do not need to prove your conjecture.
- Reflection: Think about what we have done in class so far, including what is on this exam. Describe two ways in which the material we have been considering will be useful to you now or in the future as a middle or high school teacher. Try to be as specific as possible.