Appendix F End of module info

The MATH111 end-of-module test will take place on Thursday $10^{\text{th}}$ November, starting at 4pm. Workshop groups 1–8 will sit the test in Bowland LT, while workshop groups 9–15 will sit the test in Minor Hall. Please make sure that you know which venue you are going to! This information should also be on your electronic timetable, except for students with special circumstances; they will receive an email with instructions directly from George Moran.

The test will last 40 minutes and consist of questions worth a total of 50 marks. Please bring a pen, and possibly a calculator; if you bring a calculator which has a permanent memory, the memory must be cleared. Paper will be supplied.

The test paper will consist of

1. $\bullet$

   exercises similar in nature to those known from workshops and assessments; and

2. $\bullet$

   bookwork, in the form of the statement of certain definitions and/or theorems, together with their proof in the latter case.

To help you revise for the second part, I have compiled the following guidelines (for the test only, not the June exam): all definitions are examinable; the results that I shall ask you to state and prove will be taken from the following list, always referred to by their name, not the number in the notes. (I have included the numbers here to help you identify the results when revising for the test):

1. (i)

   prove that $\sqrt{2}$ is irrational (Example 3.4.2);

2. (ii)

   state and prove the Division-with-Remainder Theorem (Theorem 4.1.8);

3. (iii)

   state and prove Bézout’s Theorem (Theorem 4.2.17);

4. (iv)

   state and prove Euclid’s Lemma (Theorem 4.4.8);

5. (v)

   state and prove a formula which expresses the lowest common multiple of a pair of natural numbers in terms of their highest common factor (see Theorem 4.5.5);

6. (vi)

   prove that there are infinitely many prime numbers (Theorem 4.7.1);

7. (vii)

   state and prove the result which characterizes when a linear congruence has solutions (Theorem 5.2.2);

8. (viii)

   define what is meant by the sum and the product of two congruence classes, and prove that these definitions make sense (see Lemma 6.3.2 and Definition 6.3.3);

9. (ix)

   show how to construct the integers from the natural numbers (as in Section 6.4).

Note that not all of the above proofs are equally long/complex; this will be reflected in the number of marks allocated to them. For some of the longer proofs, I may ask you to do only part of it; for instance, ‘‘State the Division-with-Remainder Theorem, and prove the existence part of it’’ would be a possible question of this form.