Download of the prime factorizations of both 24 and 60.

INVESTIGATION 4.3:
USING PRIME FACTORIZATIONS
Learning Targets

I can find common factors, greatest common factors,
common multiples, and least common multiples using
prime factorizations
Lets Revisit some material

What are the common factors of 24 and 60?

What are the common multiples of 24 and 60?
Prime Factorization

Using either the division method or the factor tree
method, find the prime factorization of 24 and 60.
60
10
5
x
x
6
2 3
x
2 24
2 12
2
60  5  3 2  2
2 6
3
24  3 2  2  2
Common Factors
Here we have a Double Bubble Map to find the common factors of 24
and 60. Let’s pick a common factor of 24 and 60.
1
24
2
5
10
15
3
24
8
60
20
4
6
30
12
60
24  3  2  2  2
60  5  3  2  2
12  3 2  2
What does the factorization of
12 have in common with the
factorizations of 24 and 60?
Find the prime factorization of 12 with your
strategy of choice and write the prime
factorization below the prime factorizations
of 24 and 60 without exponents.
24  3  2  2  2
60  5  3  2  2
6  3 2
What does the factorization of 6
have in common with the
factorizations of 24 and 60?
Lets pick another common factor. Find the
prime factorization of 6 with your strategy of
choice and write the prime factorization
below the prime factorizations of 24 and 60
without exponents.
Our Conjecture

Can we make any conjectures about the prime
factorization of any common factors of 24 and 60?
 The
prime factorizations of common factors are
“substrings” of the factorizations of both 24 and 60.
Common Multiples



For some kinds of problems, we need to find
common factors.
For others we need to find common multiples.
Let’s think about common multiples and see whether
we can find a way to use prime factorization to
help us find them.
Common Multiples

What are some common multiples of 24 and 60?
Common Multiples
Questions to ponder


If we find the product of two numbers, for example
24 x 60, will the product be a common multiple? If
so, why?
Is this product a multiple of both 24 and 60? How
do you know?
Common Multiples
Questions to ponder

If so, what do you multiply 24 by to get
3


Multiply by 5 x 3 x 2 x 2 or 60.
What do you multiply 60 by to get
3

x 2 x 2 x 2 x 5 x 3 x 2 x 2?
x 2 x 2 x 2 x 5 x 3 x 2 x 2?
Multiply 60 by 3 x 2 x 2 x 2 or 24.
Common Multiples


This means that both 24 and 60 “live” in the product
of the two numbers.
Can you circle prime factors from the product that
will give you 24?
24  60 : 3 2  2  2  5  3 2  2


Can you circle prime factors from the product that
will give you 60?
Let’s use these ideas to explore the questions in
Problem 4.3
Problem 4.3
You may work with a shoulder partner
1)
2)
3)
Write the prime factorizations of 72 and 120.
What is the longest string common to both
factorizations?
What is the greatest common factor of 72 and 120?
How do you know?
What is the shortest string of factors that includes
the prime factorizations of both 72 and 120? Can
you find a smaller common multiple of 72 and 120?
Why or why not?
Homework

Investigation 4.3 ACE (19-22, 32)