Download 02-23 5.2 Primes and Composites

§5.2 Primes and
Composites
2/23/17
Today We’ll Discuss
Review!
How do we decide when a number is
prime or composite?
How do we use factor trees?
How do we find and use all primes less
than 100?
Review
What are the divisibility tests for
2,3,4,5,6,8,9,10?
What about 7?
Divisibility Test for 7
Test #1: Take the digits of the number in
reverse order, from right to left,
multiplying them successively by the digits
1,3,2,6,4,5 repeating this sequence of
multipliers as long as necessary. Add the
products. If the sum is divisible by 7, then
so is the whole number!
Divisibility Test for 7
Ex: Is 1603 divisible by 7?
3(1) + 0(3) + 6(2) + 1(6) = 21
7 | 21 so yes!
Divisibility Test for 7
Test #2: Remove the last digit, double it,
subtract it from the truncated original
number and continue doing this until only
one digit remains. If this is 0 or 7, then the
original number is divisible by 7.
Divisibility Test for 7
Ex: Is 1603 divisible by 7?
160 – 2(3) = 154
15 – 2(4) = 7
Yes!
Divisibility Test for 7
a.) 1538
b.) 7861
c.) 639
d.) 749
Definitions of Prime & Composite
Definition: A natural number is a prime
number when it has exactly two distinct
factors: 1 and itself. (1 is not prime!)
Definition: A natural number is a composite
number when it has more than two distinct
factors.
Using the Area Model
Remember that the multiplication of any
two factors can be represented in an area
problem.
Asking students to take a 1 x ___ area
model and make a new one is a great way
to show when a small number is prime or
composite.
Using the Area Model
Example
1) Use the area model to show all the
different ways to factor 20.
2) Use the area model to discuss why 23 is
a prime number.
List of Primes
Let’s take a look at the prime numbers less
than 100.
Sieve of Eratosthenes!
The Fundamental Theorem of
Arithmetic
Every natural number greater than 1 is
either a prime number, or can be written
as the product of prime factors in exactly
one way (if order does not matter).
Definition: The product above is called the
prime factorization of a number.
Factor Trees
Factor trees are used to keep track of
factors in the process of finding a prime
factorization.
By the FToA, all prime factorizations of a
given number are the same, so you can
start however you like!
Example
Write the prime factorization of each
number.
a) 24
b) 126
c) 225
Example
Write the prime factorization of each
number.
a) 5929
b) 3500
c) 3773
Factor Tree Tips
It is always helpful to divide out as large of
factors as you go along. This helps to
shorten the process.
Don’t forget that if you are attempting to
pull out all prime factors, certain factors
may appear multiple times.
Testing only Prime or Composite
If you are asked only if a number is prime
or composite, you only have a bit of work
to do….
Theorem: To test for prime factors of a
number n, you need only check for prime
factors that are less than or equal to the
square root of n.
Example
Tell whether each number is prime or
composite.
a) 901
b) 223
Goldbach’s Conjecture
Conjecture: Every even integer greater
than 2 can be written as the sum of two
primes.
Homework #11 - §5.2
Pages 176-179
#4, 6, 11, 14, 18, 20, 23, 28, 38, 41