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Number Theory
Evens – divisible by 2 with no remainder, ends in 0, 2, 4, 6, or 8
Odds – ends in 1, 3, 5, 7, or 9
Factors– numbers that can be multiplied to get a number example F6 = 1, 2, 3, 6
Multiples – get multiples by multiplying
example M6 = 0, 6, 12, 18, 24, 30 ….
Prime number – number whose only factors are one and itself examples: 2, 3, 5, 7, 11, 13, 17
Composite – has more than two factors
Rules for divisibility
2 – ends in 0, 2, 4, 6, 8
3 – if sum of the digits is divisible by 3
4 – if last two digits are divisible by 4
5 – ends in 0, 5
6 – if divisible by 2 and 3
8 – if last three digits are divisible by 8
9 – if sum of the digits is divisible by 9
10 – ends in 0
Factor trees and prime factorization
60
6
2
10
3
2
5
22 (3) (5)
Factor Game
Bad first moves
30 – proper factors are 1, 2, 3, 5, 6, 10, 15 = 42
12 – proper factors are 1, 2, 3, 4, 6 = 16
These are abundant numbers – the sum of the proper factors is greater than the given number
Good first moves
29 – proper factor is 1
All primes
27 – proper factors are 1, 3, 9, = 13
16 – proper factors are 1, 2, 4, 8, = 15
25 –proper factors are 1, 5
These are deficient numbers – the sum of the proper factors in less than the given number
Even moves
6 – proper factors are 1, 2, 3 = 6
28 – proper factors are 1, 2, 4, 7, 14 = 28
These are perfect numbers – the sum of the proper factors is equal to the given number.
There are only 30 known perfect numbers. The first four are 6, 28, 496, and 8128.
Fractions
Numerator
Denominator
To change a fraction to a decimal, divide the numerator by the denominator example 3/5 = .6
Mixed fraction – whole number and fraction
example 5 1/3
Improper fraction – numerator is greater than the denominator example 23 / 5
This would reduce to 4 3/5
Rational number – any number that can be written as a fraction, includes all whole numbers, all
integers, all fractions, repeating decimals, terminating decimals
Irrational numbers - cannot be written as a fraction, examples π, √2, √3.
Any decimal that goes on forever with no pattern (like π).
Reals – The set of rational and irrational numbers.