# 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:

12 | 108 | 1284 |

1 + 2 = 3 | 1 + 0 + 8 = 9 | 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”

__ 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:

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