1. Factors: Divisibility Rules, Exponents, Prime Factorization and Greatest Common Factor (GCF) Mr. Martin

2. 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

3. 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

4. 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

5. Exponents Exponents show repeated multiplication –E.g., 4 3 = 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., x 5 = (x)(x)(x)(x)(x) –E.g., cm x cm x cm = cm 3 With numbers or variables to the second power, we often say “squared.” For example, for 4 2 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 4 3 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.

6. “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.

7. 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

8. 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

9. 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

10. 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

11. Example: Finding GCF of 54 and 144

12. Example: Finding GCF of 12, 16 and 20

13. Ex: Finding GCF of 12x 3 y and 18 x 2 y 2