3. 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.

4. How many ways can you factor 100? Let's do it together.

5. Shortcuts to finding the GCF and LCM
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
***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 = 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.
LCM = 2 x 2 x 3 x 2 x 5 = 120
Use this method to find the LCM of 125 and 80

6. Investigation 4 Notes - Answer
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
***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 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 (using 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 = 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 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.
LCM = 2 x 2 x 3 x 2 x 5 = 120
Use this method to find the LCM of 125 and 80
LCM = 5 x 5 x 5 x 2 x 2 x 2 = 1,000

14. Prime Time Problem 4.2 Answers
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 x 2 x 2 = 8 so 9 is paired with 8.
E.) 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.

15. Make a Prime Factorization Tree30
Answer
2 x x 5
2x3x5
Help
The last number on each arrow should be a prime number. Prime numbers are circled in black.
Pull
Way to Go!
Draw arrows and write the factors for the number until the factors are all prime numbers. Then Check your answers.

16. Make a Prime Factorization Tree84
Answer
2 x
6 x 7
2 x 3
2x2x3x7
Help
The last number on each arrow should be a prime number. Prime numbers are circled in black.
Pull
Way to Go!
Draw arrows and write the factors for the number until the factors are all prime numbers. Then Check your answers.

17. Make a Prime Factorization Tree63
Answer
3 x
3 x 7
3x3x7
Help
The last number on each arrow should be a prime number. Prime numbers are circled in black.
Pull
Way to Go!
Draw arrows and write the factors for the number until the factors are all prime numbers. Then Check your answers.

18. 91 x 1(IS NOT a prime #) 91 is a prime number
Make a Prime Factorization Tree 91
Answer
91 x 1(IS NOT a
prime #)
91 is a prime
number
Help
The last number on each arrow should be a prime number. Prime numbers are circled in black.
Pull
Way to Go!
Draw arrows and write the factors for the number until the factors are all prime numbers. Then Check your answers.

19. Make a Prime Factorization Tree
128
Answer
2 x x 8
2 x x4
2 x x 2
2x2x2x2x2x2x2
Help
The last number on each arrow should be a prime number. Prime numbers are circled in black.
Pull
Way to Go!
Draw arrows and write the factors for the number until the factors are all prime numbers. Then Check your answers.