Prime Numbers and GCF

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:

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”

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:


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:


Objective 4: Find the greatest common factor of a give 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 give 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 monomial

• Find the prime factorization of each monomial
• Treat the variables like prime number

• Find the G. C. F. of: