Prime Numbers and Prime Factorization

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Prime Numbers and Prime Factorization
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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 written as a product of primes 3 x 5

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To find the prime factorization

1. Divide the number by the first prime number
   possible.
2. Circle the prime number, and continue with the
   other factor.
3. Divide the new factor by a prime number.
4. 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
Both numbers are prime, leaving us with all
primes.
5 x 5
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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

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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.
• 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
2 x 2 x 3 x 5 x 7 or 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 3 x 3 x 3 or 2 x 33
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Try this on your own
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Try this on your own
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Try this on your own
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Try this on your own
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Homework Time!