MTH212

Notes on Number Theory and Fractions

Winter 2019

`b` divides `a`, `b|a`, if and only if, there is a `q` such that `a=b*q`.

for example

- `2|6` since `6=2*3`
- `6 cancel{|} 2`
- `0|2`
- `2|4n` for all whole numbers `n`

- If `d | a` and `d | b`, then `d | (a+b)`.

Ex: `3|6` and `3|9`, so `3|15` - If `d | a` and `d cancel{|} b`, then `d cancel{|} (a+b)`.

Ex: `3|6` and `3 cancel{|} 10`, so `3 cancel{|} 16` - If `d | a` and `d | b` and `a>=b`, then `d | (a-b)`.
- If `d | a` and `d cancel{|} b` and `a>=b`, then `d cancel{|} (a-b)`.
- If `d cancel{|} a` and `d | b` and `a>=b`, then `d cancel{|} (a-b)`.

- 2
- the ones digit is a `0,2,4,6,8`
- 3
- the sum of the digit is a multiple of 3
- 4
- the two right-most digits form a multiple of 4
- 5
- the ones digit is a `0,5`
- 6
- meets the tests for 2 and 3
- 8
- the three right-most digits form a multiple of 8
- 9
- the sum of the digit is a multiple of 9
- 10
- the ones digit is a `0`
- 11
- the difference between the sum of the digits from the even powers of 10 and the sum of the digits from the odd powers of 10, is a multiple of 3

Consider the image below:

The corners show divisibility by 2, 3, 5, and 7.
Numbers with any colored corners are **composite**.
Orange numbers are **prime**.

- Divisor
- a number that is a factor of another number

`b` divides `a`, `b|a`, if and only if, there is a `q` such that `a=b*q`. - Factor
- a number that is a divisor of another number
- Multiple
- the product of a given number with any whole number
- Prime
- exactly two distinct divisors
- Composite
- more than two distinct divisors
- Factorization
- write as a product of whole numbers
- Prime Factorization
- write as a product of only prime numbers
- Factor Trees
- factor a number as a pair of factors; repeat with each factor until only prime numbers remain.

Consider 24 and 18:
**24:** 1, 2, 3, 4, 6, 8, 12, 24

**18:** 1, 2, 3, 6, 9, 18

List the factors of each number.

Identify the common factors.

1, 2, 3, 6

6 is the greatest of the common factors.

Consider 24 and 18:
**`24 = 2^3*3`**

**`18 = 2*3^2`**

Find the prime factorization of each number.

Select the lowest power of each common prime to multiply.

`2*3 =6`

Consider 24 and 18:
**`24 = 18*1+6`**

**`18 = 6*3+0`**

Repeated division of a sort

SOnce you get a zero remainder, the previous remainder must be the GCF.

GCF `=6`