Primes, Factors,

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Primes, Factors, Multiples NOtes
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Learning Objectives

• Find the common factors of two or more numbers
• Determine the greatest common factor (GCF) of two
  or more numbers
• Determine whether a number is prime, composite,
  or neither.
• Determine the prime factorization of a given
  number
• Write the prime factorization using exponents
• Find the common multiples of two or more numbers
• Determine the least common multiple (LCM) of two
  or more numbers.

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Factors
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Composites Primes

• Composite Number a number that has more than two
  factors.
• Example 4, 28, 100
• Prime Number a number that only has two factors
  one and itself.
• Example 5, 29, 101
• Primes less than 40

2 3 5 7 11 13 17 19 23 29 31 37
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Prime Factorization

• Two numbers that are neither prime nor composite
  0 and 1 .
• Prime Factorization writing a number as a
  product of its prime factors.
• Example 30 2 x 3 x 5
• You find the prime factorization of a number by
  making a factor tree.

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Finding the Prime Factorization of100
STEPS Calculations
Break the number down into two of its factors, using a factor tree. 100 20 5
2. Since 5 is a prime number we circle it (this means it is one of the prime factors of 100). 20 is a composite number, we repeat Step 1. 5 4
Since 5 is a prime number we circle it. 4 is a composite number, we repeat Step 1. 2 2
Since all the numbers are broken into prime factors, we use them to write the product. 2 x 2 x 5 x 5
Then we write the prime factorization in exponential form (using exponents). 2² x 5²
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Common Factors

• Common Factors factors that two or more numbers
  have in common.
• Example Find all the common factors of 10 and 20
  by listing all the factors.
• 10 1, 2, 5, 10
• 20 1, 2, 4, 5, 10, 20
• Greatest Common Factor (GCF) the biggest factor
  that two numbers have in common.

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GCF

• There are two different ways to find the GCF of
  two or more numbers.
• Using a list List all the factors of each
  number. Circle the greatest common factor that
  appears in the list.
• 12 18
• 1 12 1 18
• 2 6 2 9
• 3 4 3 6

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GCF continued

• Using Prime Factorization Find the prime
  factorizations of each number. Circle all the
  common prime factors. Multiply the common prime
  factors to get the GCF.
• 12 18
• 4 3 3 6
• 2 2 2 3
• 2² x 3 2 x 3²
• GCF 2 x 3 6

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You use the GCF to solve problems like the
following

• Museum employees are preparing an exhibit of
  ancient coins. They have 49 copper coins and 35
  silver coins to arrange on the shelves. Each
  shelf will have the same number of copper coins
  and the same number of silver coins. How many
  shelves will the employees need for the exhibit?
• 7 shelves

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Multiples

• Multiple a product of that number and another
  whole number.
• Example
• The multiples of 8 - 8, 16, 24, 32, 40
• Common Multiples
• Example Find some common multiples of 4 and 6 by
  listing at least ten multiples
• 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44
• 6, 12, 18, 24, 30, 36, 42, 48, 53, 60, 66

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LCM

• Least Common Multiple the smallest multiple
  that two number have in common.
• There are two different ways to find the LCM of
  two or more numbers.
• Using a list List about ten multiples of each
  number. Circle the lowest common multiple that
  appears in the list.
• 10 10, 20, 30, 40, 50, 60, 70, 80, 90, 100
• 12 12, 24, 36, 48, 60, 72, 84, 96, 108

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LCM continued

• Using Prime Factorization Find the prime
  factorizations of each number. Write them in
  exponential form. Take each number that is used.
  If they are used more than once, use the one with
  the biggest exponent. Multiply the common prime
  factors to get the GCF.
• 10 12
• 5 2 3 4
• 2 2
• LCM 2² x 3 x 5 60

2 x 5
2² x 3
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You use the LCM to solve problems like the
following

• Rod helped his mom plant a vegetable garden. Rod
  planted a row every 30 minutes, and his mom
  planted a row every 20 minutes. If they started
  together, how long will it be before they both
  finish a row at the same time?
• 60 minutes (1 hour)

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