Download 4-1 Number Theory

4-1 Number Theory
Definitions:
Even: Numbers in the form of 2a, such that a is an element of N, W, or Z.
Odd: 1) Numbers in the form of 2a – 1, when a is an element of N;
2) Numbers of the form 2a + 1, when a is an element of W or Z.
Factors: A finite group of numbers in W that are all divisors of a specific number:
 Factors of 24 = {24, 12, 8, 6, 4, 3, 2, 1}
 Factors of 17 = {17, 1}
Factorization: A finite pairing of numbers in W that multiplied together yield the desired
product. Factorizations are not necessarily unique (order is not important):
 Factorizations for 24:
o 24  1; 12  2; 8  3; 6  4
 Factorization of 17:
o 17  1
Prime: Numbers in W such that they only have 1 and the number itself as factors: 17 is
prime as shown above.
Composite: Numbers in W such that they have more than 1 and themselves as factors:
24 is composite as shown above.
Prime Factors: The list of prime numbers that are the divisors (factors) of a given
number:
 Prime factors of 24: {2, 3}
 Prime factors of 17: {17}
Prime Factorization: The unique prime factors whose product results in a specific
number:
 Prime factorization of 24: 23  3
 Prime factorization of 17: 17
Fundamental Theorem of Arithmetic: The prime factorization, except for order, of any
given number is unique. The product results in one and only one distinct number.
Factor test: In order to know how many numbers must be tested for a given number,
first take the square root of the number and then test all of the factors less than
or equal to the whole number portion of the number. If looking for prime factors,
only test primes less than or equal to the whole number portion of the number:
 Factor test on 24: 24  4.898979 ; only have to test factors less than or equal to 4
to find factors or prime factors
 Factors of 17: 17  4.1231; only have to test factors less than or equal to 4 to find
factors or prime factors
Page: 100 – 108
 Even and Odd Numbers: 1 – 5
 Factors and Primes: 6 – 10
 Exercises: 1 – 10 (homework/practice)
Activity: Sieve of Eratosthenes
 100’s chart patterns
Activity: Patterns with Factors
 100’s chart patterns
 Shifted 100’s chart patterns
 More Shifted 100’s chart patterns
 Other types of number chart patterns
 More number chart patterns
 Make your own pattern