# Number Theory with Glenn Olsen

YouTube Playlist (83 vids), Published on Jul 25, 2014 \[<https://www.youtube.com/playlist?list=PLr3WmPgPWZfX1HUpeyKkP6ir2wOFhqXMO>]

## Number Theory Topics

* Subsets of Numbers
* Divisibility
* Primes
* GCD (Greatest Common Divisor)
* Euclidean Algorithm
* LCM (Least Common Multiple)
* Relationship between GCD and LCM
* Total Number of Factors
* Units Digit
* Remainder
* Trailing Zeros
* Counting Multiples
* Sum of the First n Natural Numbers
* Linear Diophantine Equations
* Modular Arithmetic
* Gear Problem
* Number Sense

## Subsets of Numbers

Subsets: conventions used (what is meant by different sets):

### Number sets:

* Integers: $$\mathbb{Z} = { \dots, -2, -1, 0, 1, 2, \dots }$$
* Whole numbers: $$\mathbb{N\_0} = {0, 1, 2, \dots}$$
* Counting numbers: $$\mathbb{N^+} = {1, 2, \dots}$$

### Parity subsets

$$
\begin{align}
Even =& { n|n=2k,   & {n,k\in\mathbb{Z}} } = {0,2,4,6\dots } \\
Odd  =& { n|n=2k+1, & {n,k\in\mathbb{Z}} } = {1,3,5,7\dots }
\end{align}
$$

### Perfect Squares

* $$f(n)=n^2$$
* sum of first n odd numbers:   &#x20;

  sum of first 7 off numbers = 1+3+5+7+9+11+13=

$$
\displaystyle
\sum\_{i=0}^{n} 2i+1 \equiv f(n)=n^2,\ n\in\mathbb{N^+}\\
$$

$$
\displaystyle
\sum\_{i=0}^{k=5} 2k+1 = \\
1 + (2+1) + (2\cdot2+1) + (3\cdot2+1) + (4\cdot2+1) = \\
1 + 3 + 5 + 7 + 9 = 25
$$

```
1 2 3 4 5  6  7  8
1                   = 1
1+3                 = 4
 " +5               = 9
  "  +7             = 16
   "   +9           = 25
1+3+5+7+9+11        = 36
1+3+5+7+9+11+13+15  = 49
```

$$
n^2 = 0^2 = 1^2 = 1 \quad n \in \mathbb{N\_0} \\
n^2 = 1^2 = 1       \quad n \in \mathbb{N^+}
$$

### Pythagorean Triples

* $$a^2 + b^2 = c^2$$
* It isa very common that two elements in a triple differ by 1 or 2
* most common: (3,4,5)
* common: (5,12,13), (8,15,17), (7,24,25)
* not so common: (20,21,29), (12,35,37), (9,40,41)

## Divisibility

Divisibility: if `a` and `b` are ints, then `a` divides `b` if `b = a*n` for some int `n`.

## Divisibility rules

The integer, $$n$$, consisting of $$k+1$$ (base 10) digits, $$d\_k \dots d\_1 d\_0$$, is divisible by $$m$$ if:

* rule 2, m=2: if the last digit is divisible by 2
* rule 4, m=4 (2^2): if the last 2 digits are divisible by 4
* rule 8, m=8 (2^3): if the last 3 digits are divisible by 8
* rule 3, m=3: if the sum of all the digits is divisible by 3
* rule 9, m=9: if the sum of all the digits is divisible by 9
* rule 5: last digit must be 0 or 5
* rule 10: last digit must be 0
* rule 6 (2\*3): if rules 2 and rule 3 both hold
* rule 12 (3\*4)

$$
\displaystyle
\sum\_{i=0}^{k} d\_i \cdot 10^i \\
n = d\_{k+1} \dots d\_1 d\_0
$$


---

# Agent Instructions: Querying This Documentation

If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://mandober.gitbook.io/math-debrief/501-number-theory/terms/number-theory2.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.