1. Fractions and Factors In order to work with fractions efficiently, it is important to understand some concepts about factors, prime numbers and composite numbers We will use these concepts to develop equivalent fractions when we need to reduce a fraction to lowest terms or change fractions to one with a common denominator.

2. Factors Factors are the numbers that are multiplied together to get a product. We can write 12 as the product of two factors in any of the following ways: 3 • 4 = 12 2 • 6 = 12 1 • 12 = 12 If asked for all of the factors of 12, the answer would be: 1, 2, 3, 4, 6, and 12

3. Finding All Factors Try using the Rainbow Method to find all factors of a given number. For example, to find all of the factors of 36: Start with 1 which is a factor of every number. Since 1 X 36 = 36, we place 1 at one end and 36 at the other. 36 1

4. Finding All Factors Try using the Rainbow Method to find all factors of a given number. For example, to find all of the factors of 36: Since 2 is a factor of 36, and 2 X 18 = 36, we place the factors 2 and 18 inside the first set of factors. 36 1 2

5. Finding All Factors Try using the Rainbow Method to find all factors of a given number. For example, to find all of the factors of 36: Since 3 is a factor of 36, and 3 X 12 = 36, we place the factors 3 and 12 inside the next set of factors 1 2 3 12 18 36

6. Finding All Factors Try using the Rainbow Method to find all factors of a given number. For example, to find all of the factors of 36: Since 4 is a factor of 36, and 4 X 9 = 36, we place the factors 4 and 9 inside the other sets of factors. 1 2 3 4 9 12 18 36

7. Finding All Factors Try using the Rainbow Method to find all factors of a given number. For example, to find all of the factors of 36: Now we try 5. But that is not a factor of 36, so we go on to 6. 6 X 6 = 36, so we include the factor 6 in our rainbow. We have already captured all of the factors greater than 6, so we are done. 1 2 3 4 6 9 12 18 36 Solution: All of the factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36

8. Prime and Composite Numbers A Prime number can only be divided by 1 and itself. The first 10 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 A Composite number is composed of more than one prime factor and can be divided by other factors, as well as 1 and itself. The first 10 composite numbers are: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18 NOTE: The number 1is considered neitherprime nor composite.

9. Prime Factorization It is sometimes necessary to be able to break a number down into its prime factors. This process is called Prime Factorization. We can use a factor tree to determine the prime factorization of a number. To determine the prime factorization of 12, we first choose any set of factors for 12, such as 3 X 4 12 / br /> 4 / br /> 2 2 3 is already a prime number, but 4 is not, So we break it down into its factors Solution: the prime factorization of 12 is 3 •2 • 2

11. Then, if the factor is prime, we circle it.

13. Reduce a Fraction to Lowest Terms Write fraction in lowest terms using the Prime Factoring method. First, write the numerator and denominator as the product of their primes. Divide out any common factors. Since 2 and 3 have no more common factors The fraction is in lowest terms.

14. Finding an Equivalent Fraction 3 Let’s say we need to rewrite — with a denominator of 15. 5 Remember that if we multiply numerator and denominator by the same number, we get an equivalent fraction. Since 5 •3 = 15, we need to multiply the numerator by 3 as well. = 1 -- So when we multiply both numerator and denominator by 3 we are multiplying the original fraction by 1.

15. Lowest Common Denominator To find the LCD (Lowest Common Denominator) for two fractions, determine the prime factorization for each denominator The LCD will include each different factor, and those factors will be used the maximum number of times it appears in any factorization To find the LCD for and , first list the prime factorization for each denominator. 3 — 20 5 — 24 20 = 2 • 2 • 5 and 24 = 2 • 2 • 2 • 3 5 appears once in the factorizations, 3 also appears once, but 2 appears at most three times, so the LCD will be 2 • 2 • 2 • 3 • 5 or 120.

16. Change to Common Denominator Once we have found the LCD for two fractions we can change them to equivalent fractions with a common denominator. Since 20 • 6 = 120 we multiply the numerator by 6 as well Since 24 • 5 = 120, we multiply the numerator by 5 as well