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 162 x 83 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?

1 x 182 x 93 x 64 x ??5 x ??6 x 3 Repeat! Cross it out!

We’re done!

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

What are the factors of 7?

1 x 72 x ??3 x ??

4 x ??5 x ??6 x ??7 x 1 This works, but it is a repeat. We are

done.

The only factors of 7 are 1,7

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: 3 x 5

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 X 50

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

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

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

2 X 25

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.

5 x 5Both numbers are prime, leaving us with all primes.

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

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

2 x 2 x 3 x 5 x 7

or

22 x 3 x 5 x 7

Another Example

420

2 x 210

2 x 105

3 x 35

5 x 7