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University of Toronto, October 18, 2004

This document in PDF: Exam.pdf

**Solve 4 of the following 5 problems. ** Each problem is
worth 25 points. If you solve more than 4 problems indicate very
clearly which ones you want graded; otherwise a random one will left
out at grading and it may be your best one! Write your answers in the
space below the problems and on the front sides of the extra pages; use
the back of the pages for scratch paper. Only work appearing on the
front side of pages will be graded. Write your name and student number
on each page. If you need more paper please ask the tutors. You have an
hour and 50 minutes.

**Allowed Material: ** Any calculating device that is not
capable of displaying text.

**Problem 1. ** Find formulas for
,
and
in terms of
. (You may use any
formula proven in class; you need to quote such formulae, though you
don't need to reprove them).

**Problem 2. **

- Let be a natural number. Prove that any natural number can be written in a unique way in the form , where and are integers and .
- We say that a natural number is ``divisible by '' if is again a natural number. Prove that is divisible by if and only if is divisible by .
- We say that a natural number is ``divisible by '' if is again a natural number. Is it true that is divisible by if and only if is divisible by ?

**Problem 3. ** A function is defined for
and has the graph plotted above.

- What are , and ?
- Let be the function . What are , and ?
- Are there any values of for which ? How many such 's are there? Roughly what are they?
- Plot the graph of the function . (The general shape of your plot should be clear and correct, though numerical details need not be precise).
- (5 points bonus, will be given only to excellent solutions and may raise your overall exam grade to 105!) Plot the graphs of the functions and .

**Problem 4. **

- Define `` '' and `` ''.
- Prove that if and then .
- Prove that if then and .
- Draw the graph of some function for which and .

**Problem 5. ** Give examples to show that the
following definitions of
do not
agree with the standard one:

- For all there is an such that if , then .
- For all there is a such that if , then .

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Dror Bar-Natan 2004-10-18