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:

  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)

$$\displaystyle \sum_{i=0}^{k} d_i \cdot 10^i \\ n = d_{k+1} \dots d_1 d_0$$