Download 3.5 Prime Numbers and GCF

Prealgebra, Chapter 3 Expressions & Polynomials: 3.5 Prime Numbers and G.C.F. Definitions:
• Prime number – A natural number other than 1 that has exactly two
different factors, 1 and the number itself
o 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, …...
•
Composite number – A natural number that has factors other than 1 and
itself
o 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, ……
•
Prime factorization – A number written as a product of only prime
numbers
•
Greatest common factor – The largest number that divides all given
numbers with no remainder
Objective 1: Determine is a number is prime, composite or neither
•
There are shortcuts in determining if a number is divisible by another
number:
o Even #’s are all divisible by “2”
o To determine if a # is divisible by “3”, add the digits up and if the
answer is equal to a number divisible by “3”, then so is the #.
Example:
12
1+2=3
108
1+0+8=9
1284
1 + 2 + 8 + 4 = 15
o Anything ending in a “0” or a “5” is divisible by “5”
o To determine if a number is divisible by “9”, follow the same rules
as you did for the # 3, but the digits must add up to a # divisible by
“9”
o Anything ending in a “0” is also divisible by “10”
Resource: Carson, T. (2009 ). Prealgebra – 3rd ed..
Prealgebra, Chapter 3 Expressions & Polynomials: 3.5 Prime Numbers and G.C.F. Objective 2: Find the prime factorization of a given number
•
To find the prime factorization of a number, use a factor tree
o Draw two branches below the number
o Place two factors that multiply to equal the given number at the
end of the two branches
o Repeat steps 1 and 2 for every composite factor
o Place all the prime factors together in a multiplication sentence
•
Examples:
24
^
45
^
9 5
^
3 3
6
^
4
^
2 3 2 2
Prime factor =
2*2*2*3
60
^
6
^
10
^
2 3 2 5
3*3*5
2*2*3*5
Objective 3: Find all possible factors of a given number
•
This is called factoring which is simply a list of all of the possible factors
of a given number
•
Examples:
24
1 * 24
2 * 12
3*8
4*6
50
1 * 50
2 * 25
5 * 10
81
1 * 81
3 * 27
9*9
Resource: Carson, T. (2009 ). Prealgebra – 3rd ed..
Prealgebra, Chapter 3 Expressions & Polynomials: 3.5 Prime Numbers and G.C.F. Objective 4: Find the greatest common factor of a given set of numbers by listing
•
To find the greatest common factor by listing
o List all the possible factors for each given number
o Search the lists for the greatest factor common to all lists
•
Find the G.C.F. of:
24 and 60
Objective 5: Find the greatest common factor of a given set of numbers using prime
factorization
•
To find the greatest common factor of a given set of numbers
o Write the prime factorizations of each number in exponential form
o Create a factorization for the G.C.F. that contains only those prime
factors common to all the factorizations
o Multiply
•
Find the G.C.F. of:
24 and 60
Objective 6: Find the greatest common factor of a set of monomials
•
•
Find the prime factorization of each monomial
Treat the variables like prime numbers
•
Find the G.C.F. of:
Resource: Carson, T. (2009 ). Prealgebra – 3rd ed..