Download PRIME F - Ms. Stern`s Classes

Name_______________________________________ Date __________________
Class ____________________
Ms. Stern
MATH PACKET: PRIME FACTORIZATION,
GREATEST COMMON FACTOR, & LEAST COMMON MULTIPLE
Prime Factorization
A prime number is a number whose only factors are itself and 1.
Examples are 2, 3, 13, and 37.
Any whole number can be written as a product of all its prime factors. This is the prime
factorization of the number.
We use factor trees to help determine prime factorization:
When you find a prime factor, circle it. That branch of the tree stops there.
Example:
Directions: You try. Find the prime factorization of each number and write in
exponential form.
1) 60 = ____________________
2) 75 = ____________________
3) 120 = ____________________
4) 288 = ____________________
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5) 136 = ____________________
6) 99 = ____________________
7) 225 = ____________________
8) 576 = ____________________
Greatest Common Factor
The greatest common factor (GCF) of one or more numbers is the greatest number that
is a factor of each number.
Example:
Find the greatest common factor of 32 and 80.
Step 1
List all the factors of each number.
Factors of 32: 1, 2, 4, 8, 16, 32
Factors of 80: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80
Step 2
Compare the lists. 16 is the greatest number that is a factor of both numbers, so 16 is
the greatest common factor of 32 and 80.
Directions: You try. What is the GCF of
1) 27 and 45
2) 18 and 24
3) 17 and 51
4) 26 and 65
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Greatest Common Factor Using Prime Factors
One way to find the GCF — especially for larger numbers — is using prime
factorization.
Example:
Find the GCF of 6300 and 1176.
Step 1
Find the prime factorization of each number.
Step 2
Find the prime factors the two have in common.
These numbers have two 2s, one 3, and one 7 in common.
Step 3
Multiply these prime factors together to find the GCF: 2 x 2 x 3 x 7 = 84.
The greatest common factor of 6300 and 1176 is 84.
Directions: You try. Use prime factorization to find the GCF of 1020 and 7605.
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Least Common Multiple
The least common multiple (LCM) of two numbers is the least number that is a multiple
of both numbers. (The least common multiple is used as the least common
denominator.)
Example:
Find the LCM of 25 and 30.
Step 1
List the first several multiples of each number.
Multiples of 25: 25, 50, 75, 100, 125, 150, 175, 200…
Multiples of 30: 30, 60, 90, 120, 150, 180, 210…
Step 2
Compare the lists. 150 is the least number that is a multiple of both 25 and 30.
Directions: You try. What is the LCM of
1) 4 and 6
2) 6 and 9
3) 15 and 27
4) 20 and 36
5) 8 and 18
6) 9 and 12
7) 15 and 24
8) 18 and 42
Remember: You can always find a common multiple (not necessarily the least) by
multiplying the two numbers together.
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Least Common Multiple Using Prime Factors
One way to find the LCM is using prime factorization.
Example:
Find the LCM of 6300 and 1176.
Step 1
Find the prime factorization of each number.
Step 2
Find the smallest product that includes all the prime factors of each number: In this
case, it would be
2x2x2x3x5x5x7x7
It includes the three 2s in 1176, the two 3s in 6300, the two 5s in 6300, and the two 7s
in 1176. However, it includes no extra prime factors.
Step 3
Multiply the numbers together: The LCM of 6300 and 1176 is
2 x 2 x 2 x 3 x 5 x 5 x 7 x 7 = 29400.
Directions: You try. Use prime factorization to find the LCM of 1020 and 7605.
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Using Greatest Common Factor and Least Common Multiple in Word Problems
Greatest Common Factor is used in word problems where you divide a number of
things into groups.
Example:
You have 32 cookies and 48 Munchkins for your friends. You want each friend to get
the same amount of each kind of snack. Among how many friends can you divide
the snacks?
Find the Greatest Common Factor:
List all the factors for each number:
32: 1, 2, 4, 8, 16, 32
48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
There are several common factors, but the greatest common factor is 16. So, you can
divide the snacks up among 16 friends.
Least Common Multiple is used in word problems where you have many groups of
different things (that is, multiples).
Example:
The old hot dog-bun problem: You are hosting a hot dog cookout. You go to Costco.
The hot dogs come in packages of 48. The buns come in packages of 32. What is
the least number of hot dogs and of buns you could buy and not have any left over
(that is, each hot dog must be matched to a bun and vice versa).
List multiples for each number. (Note, I did not say “list all the multiples.” Why not?):
32: 32, 64, 96, 128, 160, 192…
48: 48, 96, 144, 192, …
There are two common multiples so far, but the Least Common Multiple is 96. So, you
must buy 96 hot dogs and 96 buns in order not to have any left over.
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