### PrimeFactorizationLesson

```Prime Time Problem 4.1 Answers
A.)
Answers will vary, but the longest string in the puzzle is:
2 x 2 x 2 x 3 x 5 x 7.
B.)
It is not possible to find a longer string than is in the puzzle. The longest possible string is 2 x 2 x 2 x 3 x 5 x 7.
C.)
Some strings can be broken down further to get longer strings. For example 420 x 2 can be split up into 210 x 2 x 2.
D.)
If all the numbers in a string are prime, the string is the longest possible.
E.)
Except for order, every whole number has exactly one longest string.
How many ways can you factor 100? Let's do it together.
Investigation 4 Notes
VOCABULARY:
Fundamental Theorem of Arithmetic - a whole number can be ________,
except for order, into a _____________ of ___________ in exactly ____ way.
for example, 100 can be written as: ______________________
factorization - a string of ________________
for example, one factorization for 100 is: ______________________
prime factorization - factor string of ________ numbers
for example, the prime factorization for 100 is: ___________________
factor tree - an orderly record of your steps to find __________ _______________
Ex.
100
2
100
50
25
100
4
10
10
***Although you arrive at the final product in different ways, all have the same prime factorization. Except for order, there is only ONE
way to write it.
exponents: small raised numbers used to tell how many times a _________ is repeated.
For example, the prime factorization for 100 can be written using exponents as follows: ________________________ (this is called the
short-cut method)
Shortcuts to finding the GCF and LCM
Greatest Common Factor (GCF) - The product of the longest prime factorization string that both numbers have in common.
For ex.
24 = 2 x 2 x 2 x 3
60 = 2 x 2 x 3 x 5
GCF = 2 x 2 x 3 = 12
Use this method to find the GCF of 125 and 80
Least Common Multiple (LCM) - The product of the shortest prime factorization string that both numbers have in common.
24 = 2 x 2 x 2 x 3
LCM = 2 x 2 x 3 x 2 x 5 = 120
Use this method to find the LCM of 125 and 80
60 = 2 x 2 x 3 x 5
VOCABULARY:
Fundamental Theorem of Arithmetic - a whole number can be factored,
except for order, into a product of primes in exactly one way.
for example, 100 can be written as: 2 x 2 x 5 x 5
factorization - a string of factors
for example, one factorization for 100 is: 2 x 25 x 2
prime factorization - factor string of prime numbers
for example, the prime factorization for 100 is: 2 x 2 x 5 x 5
factor tree - an orderly record of your steps to find prime factorization
Ex.
100
2
100
50
2
25
25
5
5
100
4
5
2
10
2
2
10
5
2
5
5
***Although you arrive at the final product in different ways, all have the same prime factorization. Except for order, there is only ONE w
to write it.
exponents: small raised numbers used to tell how many times a factor is repeated.
For example, the prime factorization for 100 can be written using exponents as follows: 2² x 5² (this is called the short-cut method (usin
exponential notation)
Shortcuts to finding the GCF and LCM
Greatest Common Factor (GCF) - The product of the longest prime factorization string that both numbers have in common.
For ex.
24 = 2 x 2 x 2 x 3
60 = 2 x 2 x 3 x 5
GCF = 2 x 2 x 3 = 12
Use this method to find the GCF of 125 and 80
125 = 5 x 5 x 5
80 = 5 x 2 x 2 x 2
GCF = 5
Least Common Multiple (LCM) - The product of the shortest prime factorization string that both numbers have in common.
24 = 2 x 2 x 2 x 3
60 = 2 x 2 x 3 x 5
LCM = 2 x 2 x 3 x 2 x 5 = 120
Use this method to find the LCM of 125 and 80
125 = 5 x 5 x 5
80 = 5 x 2 x 2 x 2
LCM = 5 x 5 x 5 x 2 x 2 x 2 = 1,000
A.)
From the pictures that we draw, we can read that 100 =
2 x 2 x 5 x 5. This string is a factorization of 100 into prime numbers. We now know (from Problem 4.1) that there is only 1 prime
factorization for a number. Therefore, we call it the factorization instead of a factorization.
B.)
72 = 2 x 2 x 2 x 3 x 3
120 = 2 x 2 x 2 x 3 x 5
600 = 2 x 2 x 2 x 3 x 5 x 5
C.)
72 = 2³x 3²
120 = 2³x 3 x 5
600 = 2³x 3 x 5²
D1.) Answer may vary - example will use 9.
72 = 3 x 3 x 2 x 2 x 2
D2.) You could circle the other factors left in the factorization.
72 = 3 x 3 x 2 x 2 x 2
E.)
2 x 2 x 2 = 8 so 9 is paired with 8.
Many possible answers. Example will use the multiple 144.
Prime factorization of 72 = 3 x 3 x 2 x 2 x 2
Prime factorization of 144 = 3 x 3 x 2 x 2 x 2 x 2
The only difference is another factor of 2 is included because 144 is the second multiple of 72. Accordingly, every prime factorization for a
multiple of 72 will contain the prime factorization of 72 in it.
30
Make a Prime Factorization Tree
Draw arrows and write the factors for the number until the
Pull
Hel
p
The last number on each
arrow should be a prime
number. Prime numbers
are circled in black.
84
Make a Prime Factorization Tree
Draw arrows and write the factors for the number until the
Pull
Hel
p
The last number on each
arrow should be a prime
number. Prime numbers
are circled in black.
Make a Prime Factorization Tree
63
Pull
Hel
p
The last number on each
arrow should be a prime
number. Prime numbers
are
circled
in
black.
Draw arrows and write the factors for the number until the
91
Make a Prime Factorization Tree
Draw arrows and write the factors for the number until the
Pull
Hel
p
The last number on each
arrow should be a prime
number. Prime numbers
are circled in black.
128
Draw arrows and write the factors for the number
until the factors are all prime numbers. Then