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Prime Numbers and Prime Factorization

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

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

What are the factors of 16? 1 x 16 2x8 3 x ?? 4x4

3 is not a factor, so cross it out

doubles = done

The factors of 16 are 1,2,4,8,16

What are the factors of 18? 1 2 3 4 5 6

x x x x x x

18 9 6 ?? ?? 3

The factors are 1,2,3,6,9,18

Repeat! Cross it out! We’re done!

What are the factors of 7? 1x7 2 x ?? 3 x ?? 4 5 6 7

x x x x

?? ?? ?? 1

The only factors of 7 are 1,7

This works, but it is a repeat. We are done.

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

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

Example

•15 is a composite number. •It can be written as a product of primes: 3x5

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.

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…

Example: Prime Factorization of 100.

100 2 is a prime number, so we are done with it.

100 ÷ 2 = 50. Two is the first prime number that goes into 100.

2 X 50

Now we deal with the 50. Divide it by 2 to get the next factors.

2 X 25 Both numbers are prime, leaving us with all primes.

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.

5x5

What’s the Answer? • Now, we just list our factors with multiplication signs between them. Use the circled prime numbers. • 2x2x5x5

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 • 2 x 2 can be expressed in exponent form: 2 2 • 5 x 5 can be expressed as 5 • Put it together, and 2 x 2 x 5 x 5 is more simply put as 2 2 2 x5

Another Example 420

2 x 210 2 x 105 2x2x3x5x7 3 x 35 or 5 x 7 2 2 x3x5x7

Try this on your own: 54

Answer: 2x3x3x3 or 2 x 3

3

Try this on your own:

Try this on your own:

Try this on your own:

Try this on your own:

Homework Time!

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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

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

What are the factors of 16? 1 x 16 2x8 3 x ?? 4x4

3 is not a factor, so cross it out

doubles = done

The factors of 16 are 1,2,4,8,16

What are the factors of 18? 1 2 3 4 5 6

x x x x x x

18 9 6 ?? ?? 3

The factors are 1,2,3,6,9,18

Repeat! Cross it out! We’re done!

What are the factors of 7? 1x7 2 x ?? 3 x ?? 4 5 6 7

x x x x

?? ?? ?? 1

The only factors of 7 are 1,7

This works, but it is a repeat. We are done.

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

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

Example

•15 is a composite number. •It can be written as a product of primes: 3x5

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.

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…

Example: Prime Factorization of 100.

100 2 is a prime number, so we are done with it.

100 ÷ 2 = 50. Two is the first prime number that goes into 100.

2 X 50

Now we deal with the 50. Divide it by 2 to get the next factors.

2 X 25 Both numbers are prime, leaving us with all primes.

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.

5x5

What’s the Answer? • Now, we just list our factors with multiplication signs between them. Use the circled prime numbers. • 2x2x5x5

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 • 2 x 2 can be expressed in exponent form: 2 2 • 5 x 5 can be expressed as 5 • Put it together, and 2 x 2 x 5 x 5 is more simply put as 2 2 2 x5

Another Example 420

2 x 210 2 x 105 2x2x3x5x7 3 x 35 or 5 x 7 2 2 x3x5x7

Try this on your own: 54

Answer: 2x3x3x3 or 2 x 3

3

Try this on your own:

Try this on your own:

Try this on your own:

Try this on your own:

Homework Time!