Divisibility and Mental Math

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LESSON 4-1 Warm Up
Lesson 4-1 Warm Up
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LESSON 4-1 Warm Up
Lesson 4-1 Warm Up
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Prime Factorization (4-1)

• factors the numbers you multiply together to get
  a product.
• Example the product 24 has several factors.
• 24 1 x 24
• 24 2 x 12
• 24 3 x 8
• 24 4 x 6
• The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24
• To find the factors of a number
• Start with 1 times the number.
• Try 2, 3, 4, etc.
• If you get doubles (such as 4 x 4), then youre
  done. Repeats or doubles let you know youre
  done.
• Example What are the factors of 16?
• 3
  isnt a factor Z (doesnt go into 16), so cross
  it out
• Doubles or repeats mean your done!
• The factors of 16 are 1, 2, 4, 8, and 16.

• What is a factor?
• How do you find the factors of a number?

1 x 16
2 x 8
3 x ?
4 x 4
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Prime Factorization
LESSON 4-1
Additional Examples
You have 35 vegetable seeds to
plant in your garden. The seeds must be planted
in rows of equal length. How many seeds can be in
each row?
Look for pairs of factors of 35 to find the
possible numbers of seeds in each row.
There can be 5 row of 7 seeds in each row or 7
rows of 5 seeds in each row.
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Prime Factorization (4-1)

• prime number numbers that only have two factors
  one, and the number
• itself
• Examples 3, 5, 7, 11, 31
• composite numbers numbers that have more than
  two factors
• Examples 6, 15, 18, 30, 100
• prime factorization when a composite number is
  expressed as the product of prime numbers only
• Example 18 can be expressed as 3 x 3 x 2
• Example 40 can be expressed as 2 x 2 x 2 x 5

• What are prime numbers?
• What are composite numbers?
• What is prime factorization?

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Prime Factorization
LESSON 4-1
Additional Examples
Is each number prime or composite? Explain.
a. 61
61 has only two factors, 1 and 61. So 61 is
prime.
b. 65
Since 65 is divisible by 5, it has more than two
factors. So 65 is composite.
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Prime Factorization (4-1)

• How do you find the prime factorization of a
  number?

• To find the prime factorization of a number, make
  a factor tree as follows.
• Write the product of a prime and composite number
  under the original number and draw lines
  connecting the factors with the original number
• Circle the prime number, and repeat step 1 with
  the composite factor.
• Continue this process until the only numbers you
  have left are prime numbers.
• Multiply all of the circled numbers together.
• Example What is the prime factorization of 100?

100
2 is a prime numbers, so we are done with it.
2 x 50
2 x 25
5 is a prime numbers, so we are done with it.
5 x 5
So, the prime factorization of 100 is 2 x 2 x 5
x 5.
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Prime Factorization (4-1)

• How can we express prime factorization with
  exponents?

• Since exponents show repeated multiplication
  (i.e. 34 means 3 x 3 x 3 x 3), write any
  repeated prime numbers once and use an exponent
  to tell how many times that multiplication is
  repeated.
• Example In the previous example, we found the
  prime factorization of 100 as being 2 x 2 x 5 x
  5.
• 2 x 2 can be expressed in exponent form as 22
• 5 x 5 can be expressed in exponent form as 52
• So, 2 x 2 x 5 x 5 is more simply put as 22 x 52

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Prime Factorization
LESSON 4-1
Additional Examples
Find the prime factorization of 90.
Use a factor tree. Because the sum of the digits
of 90 is 9, 90 is divisible by 3. Begin the
factor tree with 3 30.
The prime factorization of 90 is 2 3 3 5 or
2 32 5.
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Prime Factorization
LESSON 4-1
Lesson Quiz
Write the prime factorization for each
number. 1. 36 2. 150 3. 99 4. 225
22 32
2 3 52
32 11
32 52