User:Marty/UCSC Math 110 Fall 2008/Final guide

Here are some suggestions for studying for the final.

=About the final=

The final exam has three hours scheduled. I (Marty) will attempt to write a final that takes about two hours. Without a doubt, some students will take three hours to complete and check every problem. Also, different students work faster and slower -- this does not necessarily reflect mathematical ability. I am hopeful that time pressure will not be an issue on the final exam.

The final exam will contain the following types of problems:
 * Quick "do you know it" questions.
 * These are questions like "What number between 10 and 20 is congruent to 3, mod 11?"
 * You either know how to answer these questions or you don't.
 * They are not meant to take much time, but work slowly and methodically.
 * These questions require that you know all the definitions very well.
 * These questions will comprise about 20 percent of the final.
 * Computational questions.
 * These are questions like "Find two integers $x$ and $y$ such that $37 x + 23 y = 2$".
 * These problems require computational skill, organization, and efficiency.
 * These problems take different people different amounts of time.
 * Organized writing will help, and yield partial credit too!
 * These questions will comprise about 50 percent of the final.
 * Proof questions.
 * These are questions like "Prove that a number is divisible by $4$ if its last two digits are divisible by $4$".
 * It can be very difficult to write proofs in an exam environment.
 * But, you should be able to begin a proof correctly.
 * State the hypotheses carefully!
 * Make it clear that you understand the definitions.
 * Any true, clear, and relevant statement will earn partial credit.
 * Proof questions will comprise about 30 percent of the final.

Some formulas will be given to you on the final exam. These include:
 * The formula for the discriminant of a binary quadratic form (both $b^2 - 4ac$ and $(u-v)^2 - ef$ on the topograph).
 * The formulas for Legendre symbols involving $-1$, $2$, and quadratic reciprocity.

=Preparation=

How should you prepare for the exam? You should definitely go through homework assignments with your peers and make sure that you understand every solution. You should read the material from this wiki: the "week-by-week" tab displays links to things like "Def/Factorial" (the definition of the word factorial), and "State/Zolotarevs Lemma" (the statement of Zolotarev's lemma). Read the definitions and statements; follow the links if you don't understand a word along the way. It is probably too late to learn new concepts, but it is not too late to focus on the precise definitions behind these concepts.

=Activities=

How can you practice computations for the exam? Here are some ideas:

Canonical decompositions and totients
Here is an exercise that you can practice with:
 * 1)  Choose a number between $100$ and $200$.  (For example, choose $100$).
 * 2)  Write down its canonical decomposition.  (For example, $100 = 2^2 \cdot 3^0 \cdot 5^2 \cdot 7^0 \cdots$).
 * 3)  Is the number prime or composite?  ($100$ is composite).
 * 4)  Is the number a perfect square?  (Yes.  Observe that every exponent in the canonical decomposition is even).
 * 5)  Compute $\phi$ of the number:  ($\phi(100) = \phi(2^2) \phi(5^2) = (2^2 - 2^1)(5^2 - 5^1) = (2)(20) = 40$).
 * 6) * This uses the results of State/Computing the totient of a prime power

Euclidean algorithm
Here is a Euclidean algorithm exercise:
 * 1)  Choose two numbers $a$ and $m$ between $50$ and $100$.
 * 2)  Make sure that $GCD(a,m) = 1$, by making sure that no prime number divides both $a$ and $m$.
 * 3) * Which prime numbers do you have to check?
 * 4)  Use the Euclidean algorithm to prove that $GCD(a,m) = 1$.
 * 5)  Run the Euclidean algorithm backwards to solve the Diophantine equation $ax + my = 1$.
 * 6)  Modify your solution to solve the Diophantine equation $ax + my = 3$.
 * 7)  Find a residue, $\bar x$, mod $m$, such that $\bar a \bar x = \bar 2$, modulo $m$.
 * 8)  Find a lax basis, which contains $\pm (a,m)$ as a primitive lax vector.
 * 9)  How are the previous three problems similar?  How are they different?

Quadratic forms
Here is a quadratic form exercise:
 * 1)  Choose an integer $\Delta$, such that $\Delta$ is congruent to $0$ or $1$, mod $4$.
 * 2) * Try positive integers, negative integers, perfect squares, and non-squares, for $\Delta$.
 * 3)  Invent a quadratic form $Q(x,y) = ax^2 + bxy + cy^2$ such that $b^2 - 4ac = \Delta$.
 * 4)  What do you expect the topograph to have?  Write down the features (before making the topograph).
 * 5) * Do you expect rivers, lakes, a well, what else?
 * 6)  Draw the range topograph of $Q$.
 * 7)  How many solutions are there to the Diophantine equation $Q(x,y) = 1$?
 * 8)  How many solutions are there to the Diophantine equation $Q(x,y) = 18$?

Modular arithmetic
Here are some modular arithmetic exercises:
 * 1)  Choose a prime number $p$, between $30$ and $50$.
 * 2)  Choose a prime number $q$ between $5$ and $17$.
 * 3)  Choose an integer $a$ between $2$ and $q-1$.
 * 4)  Evaluate $\bar a$, $\bar a^2$, $\bar a^3$, $\bar a^4$, etc.., modulo $q$.  What pattern do you find?
 * 5)  How is this pattern consistent with Fermat's little theorem?
 * 6)  Is $\bar a$ a quadratic residue, modulo $q$?  Check directly by squaring all of the numbers from $0$ to $q-1$.
 * 7)  Is $\bar a^{(q-1)/2}$ equal to $\bar 1$, $- \bar 1$, or $\bar 0$?  How is this consistent with your previous result?
 * 8)  Draw a graph of the "multiplication by $\bar a$, modulo $q$" function.  Compute the sign of $\mu_{\bar a}$?
 * 9)  How is this consistent with your previous results?
 * 10)  What is $\left( \frac{a}{q} \right)$?  Do you know the definition, and the relevance to the previous results?
 * 11)  Review the definitions at the page Def/Legendre_symbol.
 * 12)  Is $p$ congruent to $3$, mod $4$?  Is $q$ congruent to $3$, mod $4$?  Can you compute this quickly?
 * 13)  Compute the Legendre symbol $\left( \frac{q}{p} \right)$.
 * 14)  Do there exist integers $x,y$ such that $x^2 = q + py$?  How is this related to the previous result?