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 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. What are the factors of 16? 1 2
3 4 x x x x 16 8 ?? 4 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! Were done! What are the factors of 7? 1 2 3 4 5 6 7
x x x x x x x 7 ?? ?? ?? ??
?? 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 expressed 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
Whats the Answer? Now, we just list our factors with multiplication signs between them. Use the circled prime numbers. 2x2x5x5 We have listed 100 as a product of prime numbers. 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 420 2 x 210 2 x 105 2
2 x3x5x7 3 x 35 5x7 Try this on your own: 54 Answer: 2x3 3 Homework Time!