## Math Club Number Theory Training Session

These are some questions that I prepared for Math Club. The problems follow Paul Zeitz’s excellent book The Art and Craft of Problem Solving. You can find this hand-out is here:

https://pgadey.ca/teaching/2019-math-club/number-theory-training-talk/number-theory-training-talk.pdf (tex)

**1. Advice and Suggestions **

- Try out lots of examples.
- The small numbers are your friends.

**2. Facts and Questions **

Fact 1If we write for the statement “ divides .”

Formally, means for .

Question 2What is the largest such that is divisible by ? Idea: Find a factorization where is a small constant.

Fact 3The “divisors” of are all such that . We say is “prime” if its divisors are . We say that is “composite” if it is not prime.

Fact 4 (Fundamental Theorem of Arithmetic)Any natural number is a product of a unique list of primes.

Question 5Show that is irrational. Generalize!

Question 6Show that there are infinitely many primes. Euclid’s idea: Suppose there are finitely many and consider .

Question 7Show that there are arbitrarily large gaps between primes. That is, show that for any there are consecutive numbers which are all composite.

Question 8 (Germany 1995)Consider the sequence and . Show that this sequence contains infinitely many composite numbers.

**3. Congruence **

Fact 9 (The Division Algorithm)For any there is a unique pair such that and .

Fact 10We write if . For any there is \mbox{} such that . We say that “ is congruent to modulo ”. Congruence preserves the usual rules of arithmetic regarding addition and multiplication.

Question 11Suppose that has digits in decimal notation.

- Show that .
- Show that .
- Show that .
- Show that .

Question 12What are the last two digits of ?

Question 13Show that any perfect square is congruent to or . Conclude that no element of is a perfect square.

Question 14Show that 3 never divides .

**4. The Euclidean Algorithm **

Fact 15The “greatest common divisor” of and is:

Question 16Show that where and is the unique pair of numbers given by the division algorithm.

Question 17The Fibonacci numbers are defined so that , and for . Show that .

The Fibonacci numbers have the following curious property: Consecutive Fibonacci numbers are the worst-case scenario for the Euclidean Algorithm. In 1844, Gabriel Lamé showed: If then the Euclidean algorithm takes at most steps to calculate . Check out this great write-up at Cut the Knot.

** 4.1. Parity **

Question 18Suppose that is odd and is a permutation. Show that the number

must be even.

Question 19A room starts empty. Every minute, either one person enters or two people leave. Can the room contain people after minutes?

Idea: Consider the “mod-3 parity” of room population.

**5. Contest Problems **

Question 20Show that is not an integer for any .

Idea: Consider the largest power . Divide out by this largest power. This will make all of the denominators odd. (In fancy number theory terms, you’re using a 2-adic valuation.)

Question 21 (Rochester 2012)Consider the positive integers less than or equal to one trillion, i.e. . Prove that less than a tenth of them can be expressed in the form where , , and are positive integers.

Idea: None of , , or can be very big. For example, .

Question 22 (Rochester 2003)An -digit number is “-transposable” if and . For example, is -transposable. Show that there are two 6-digit numbers which are 3-transposable and find them.

\noindent Big Idea: Consider repeating decimal expansions.

Observe that .

Find a number with a repeating decimal of length six.

Question 23Suppose that you write the numbers on the blackboard. You now proceed as follows: pick two numbers and , erase them from the board, and replace them with . Continue until there is a single number left. Does this number depend on the choices you made?

## Canada Math Camp — Storer Calculus

The handout for the talk is available here:

https://pgadey.ca/teaching/talks/cmc-2018-storer-calculus.pdf

## Homework #5 Question 4

Consider a solid ball of radius . Cut a cylindrical hole, through the center of the ball, such that the remaining body has height . Call this the donut . Use Cavalieri’s principle to calculate the volume of . Calculate the volumes of and .

Several students have asked what looks like. Here are some pictures that I found to illustrate the concept. The donut is the region between the red sphere and blue cylinder. The golden balls below show various views of the donut. The donut should fit between the two planes and , so that it has total height .

## Malin Christersson’s Cube Toy

I was looking through the Geogebra site and found this lovely applet Orthographic Projection by Malin Christersson.

This is a lovely tool for investigating one of my favourite facts about hexagons:

*The area maximizing orthogonal projection of a cube is the regular hexagon*.

It turns out that Malin has tonnes of awesome geometry stuff online!

- Hyperbolic tiler!
- Pythagora’s Tree!
- A whole Tumblr full of great animations!

Awesome math art!

## Public Talks for UTSC

## MAT 134 — Post-Term Test #1 Survey

Thank you for filling out MAT 134 Post-Term Test #1 Survey.

Here is what Parker learned about the class!

## String Figure Poster Sketch

We’re almost ready for our final string figure presentation on May 2nd.

Now we just have to put it all together!

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