Prime Numbers and Prime Factorization

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Prime Numbers and Prime Factorization

• Lesson 3-2

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Factors

• Factors are the numbers you multiply together to
  get a product.
• For example, the product 24 has several factors.
• 24 1 x 24
• 24 2 x 12
• 24 3 x 8
• 24 4 x 6
• SO, the factors are 1, 2, 3, 4, 6, 8, 12, 24

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Finding Factors

• Start with 1 times the number.
• Try 2, 3, 4, etc.
• When you repeat your factors, cross out the
  repeat - youre done at this point.
• If you get doubles (such as 4 x 4), then youre
  done. Repeats or doubles let you know youre
  done.

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What are the factors of 16?
1 x 16
2 x 8
3 x ??
3 is not a factor, so cross it out
4 x 4
doubles done
The factors of 16 are 1,2,4,8,16
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What are the factors of 18?
1 x 18
The factors are 1,2,3,6,9,18
2 x 9
3 x 6
4 x ??
5 x ??
6 x 3
Repeat! Cross it out! Were done!
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What are the factors of 7?
The only factors of 7 are 1,7
1 x 7
2 x ??
3 x ??
4 x ??
5 x ??
6 x ??
7 x 1
This works, but it is a repeat. We are done.
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Prime and Composite Numbers

• Prime numbers are
• numbers that only have
• two factors one, and the
• number itself.
• EXAMPLES
• 3, 5, 7, 11, 31

• Composite numbers
• have more than two
• factors.
• EXAMPLES
• 6, 15, 18, 30, 100

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A Product of Primes

• Every composite number can be expressed as a
  product of prime numbers.
• This is called prime factorization.

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Example

• 15 is a composite number.
• It can be expressed as a product of primes 3 x 5

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To find the prime factorization

• Divide the number by the first prime number
  possible.
• Circle the prime number, and continue with the
  other factor.
• Divide the new factor by a prime number.
• Continue this process until the only numbers you
  have left are prime numbers.

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Remember the Prime Number List

• 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41,
  43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

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Example Prime Factorization of 100.
100
100 2 50. Two is the first prime number that
goes into 100.
Now we deal with the 50. Divide it by 2 to get
the next factors.
2 is a prime number, so we are done with it.
2 X 50
25 is not divisible by the first prime, 2. The
next prime, 3, does not work either. We must
divide by 5 to get a factor.
2 X 25
5 x 5
Both numbers are prime, leaving us with all
primes.
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Whats the Answer?

• Now, we just list our factors with multiplication
  signs between them. Use the circled prime
  numbers.
• 2 x 2 x 5 x 5
• We have listed 100 as a product of prime numbers.

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Exponent Form

• We have just listed our prime factorization for
  100 as being 2 x 2 x 5 x 5. This is repeated
  multiplication. Repeated multiplication can be
  expressed with exponents.
• Our prime numbers are our bases. The number of
  times the prime number is written is the
  exponent.
• 2 x 2 can be expressed in exponent form 22
• 5 x 5 can be expressed as 52
• Put it together, and 2 x 2 x 5 x 5 is more simply
  put as
• 22 x 52

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Another Example
420
2 x 210
2 x 105
22 x 3 x 5 x 7
3 x 35
5 x 7
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Try this on your own
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Answer
2 x 33
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Homework Time!