Prime numbers and prime factorization
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Prime Numbers and Prime Factorization. Lesson 3-2. 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.

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Prime Numbers and Prime Factorization

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Prime numbers and prime factorization

Prime Numbers and Prime Factorization

Lesson 3-2


Factors

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

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

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


What are the factors of 18

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! We’re done!


What are the factors of 7

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.


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

Prime and Composite Numbers


A product of primes

A Product of Primes

  • Every composite number can be expressed as a product of prime numbers.

  • This is called prime factorization.


Example

Example

  • 15 is a composite number.

  • It can be expressed as a product of primes: 3 x 5


To find the prime factorization

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.


Remember the prime number list

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

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.


What s the answer

What’s 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.


Exponent form

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


Another example

Another Example

420

2 x 210

2 x 105

22 x 3 x 5 x 7

3 x 35

5 x 7


Try this on your own

Try this on your own:

54

Answer:

2 x 33


Homework time

Homework Time!


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