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