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.

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

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.

1 36

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.

1 2 18 36

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

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

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 9 12 18 366

Solution: All of the factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36

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 1 is considered neither prime nor

composite.

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 / \3 4 / \ 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

Prime Factorization

• Another example: Let’s define 120 as a product of its prime factors (find the prime factorization of 120)

• First, we find any two factors of 120.

• Then, if the factor is prime, we circle it.• If not prime, we factor again and circle the primes

• Continue until we only have primes.

Solution: the prime factorization of 120 is 2 • 2 • 2 • 3 • 5

Reduce a Fraction to Lowest Terms

Write fraction in lowest terms using the Prime Factoring method.

4

6

32

22

6

4

First, write the numerator and denominator as the product of their primes.

3

2

32

22

6

4

Divide out any common factors.

Since 2 and 3 have no more common factorsThe fraction is in lowest terms.

• Let’s say we need to rewrite — with a denominator of 15.

Finding an Equivalent Fraction

53

Since 5 •3 = 15, we need to multiply thenumerator by 3 as well.

Remember that if we multiply numeratorand denominator by the same number, we get an equivalent fraction.

= 1 -- So when we multiply both numerator and denominator by 3 we are multiplying the original fraction by 1.

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.

—320

—245

20 = 2 • 2 • 5 and 24 = 2 • 2 • 2 • 35 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.

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