Factors: Divisibility Rules, Exponents, Prime Factorization and Greatest

Common Factor (GCF)Mr. Martin

Divisibility definitions• Definition divisibility – divide one

integer by another with no remainder– E.g. 6 is divisible by 3 since 6 ÷ 3 = 2

• Even numbers – end in 0, 2, 4, 6, 8– i.e. divisible by 2

• Odd numbers – end in 1, 3, 5, 7, 9– i.e. not divisible by 2

Divisibility Rules• An integer is divisible by:

– 2 if it ends in 0, 2, 4, 6, 8 (i.e., it’s even)– 3 if the sum of the digits is divisible by 3

• E.g., 342 is divisible by 3 since 3 + 4 + 2 = 9 which is divisible by 3

– 4 if the last two digits are divisible by 4• E.g., 134524 is divisible by 4 since the last two digits, 24,

are divisible by 4

– 5 if the last digit is 0 or 5– 6 if the integer is divisible by 2 and 3– 9 if the sum of the digits is divisible by 9

• E.g., 81 is divisible by 9 since 8 + 1 = 9 which is divisible by 9

– 10 if the last digit is 0

Factors

• Definition Factor – an integer A is a factor of another integer B if B ÷ A leaves no remainder– E.g., 2 is a factor of 6 since 6 ÷ 2 = 3 with no

remainder– 2 and 3 are factors of 6 since 2 x 3 = 6

• List all the factors of 36– 1, 2, 3, 4, 6, 9,1 2,18, 36 since 1 x 36, 2 x 18,

3 x 12, 4 x 9, and 6 x 6 all equal 36

Exponents• Exponents show repeated multiplication

– E.g., 43 = 4 x 4 x 4 = 64• 4 is called the base and 3 is called the exponent• We read this “4 to the third power” or “4 to the power of 3”

– E.g., x5 = (x)(x)(x)(x)(x)– E.g., cm x cm x cm = cm3

• With numbers or variables to the second power, we often say “squared.” For example, for 42 we can say “4 to the second power” or “4 squared.”

• With numbers or variables to the third power, we often say “cubed.” For example, for 43 we can say “4 to the third power” or “4 cubed.”

• How do you think the terms “squared” and “cubed” came about? Think about area and volume.

“Please excuse my dear Aunt Sally.”• We can remember the proper order of operations by

the sentence, “Please excuse my dear Aunt Sally,” or “PEMDAS.”

• It stands for “Parenthesis, Exponents, Multiplication or Division (whichever occurs first), and Addition or Subtraction (whichever occurs first).

• E.g., Simplify 6(4 + 3)2. First, do the operation within the parenthesis. We get 6(7)2. Second, do the exponent. Since 7 x 7 = 49, we get 6(49). Now multiply 6(49) = 294.– BTW: I multiplied 6(49) in my head by using the distributive

property. 6(50 – 1) = 6(50) – 6(1) = 300 – 6 = 294.

Prime and Composite Numbers

• Prime – exactly two factors; itself and one

• Composite – more than two factors

• 0 and 1 are neither prime nor composite 1 has one factor 0 really has infinite factors (0 times any

number is zero) and is treated as a special case

Prime Factorization• Prime factorization – expressing a number as the product of its

prime factors– Usually done using a factor tree– Write final factors in increasing order from right to left– Use exponents for repeated factors

Greatest Common Factor (GCF)

• Factors of: 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 24: 1, 2, 3, 4, 6, 8, 12, 24 Common factors are 1, 2, 3, 4, 6, 12 The Greatest Common Factor (GCF) on 24 and 36

is 12

• We will use the GCF later to simplify fractions in one step

Finding Greatest Common Factor (GCF)

• Do factor tree for each number

• List prime factors in order for each number

• Circle common factors

• Multiply common factors together (once, not twice)

• When listing common factor with exponents, you can just use the one with the lower exponent

• See example next page

Example: Finding GCF of 54 and 144

Example: Finding GCF of 12, 16 and 20

Ex: Finding GCF of 12x3y and 18 x2y2