> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/501-number-theory/terms/number-theory2.md).

# 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)

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