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