1) The document discusses factors, multiples, greatest common factors (GCF), and least common multiples (LCM) of numbers. 2) A number B is a factor of A if A can be divided by B with no remainder. The GCF is the greatest number that is a factor of both numbers. 3) The LCM is the lowest number that is a multiple of both numbers. To find the LCM, list the multiples of each number and take the lowest common multiple.

1. Chapter 8
Factors & Multiplication
Michelle LiSanti and Kate Muller

2. Factors 8.1
 If A and B are counting numbers and A/B equals a counting
number with no remainder than A is divisible/ evenly
divisible by B.
 The reverse of this operation is B divided by A.
 EX: 12 is divisible 4 just as 5 divides into 30
 If A and B are counting numbers, than B is a factor or a
divisor of A if*** and only if there is a counting number C.
 So A = B x C.

3. Problem 1
 7 is a factor of 21 so therefore what number times 7 equals 21
21 = 7 x 3
 This equation also states that 3, 1, and 21 are also factors of
21
 ***Divisibility and Factor are similar concepts: B is a factor
of A exactly when A is divisible by B.

4. Multiples 8.1
 If A and B are counting numbers, than A is a multiple of B if there is a
counting number C
 A = B x C  44 is a multiple of 11 due to the equation 44 = 11 x 4.
 A is a multiple of B exactly when B is a factor of A.
 Summary ofA, B, and C of counting numbers ifA = B x C
-A is a multiple of B
-A is a multiple of C
-B is a factor of A
-C is a factor of A
 **By having students make a multiplication/times table it is easier for them to
grasp the concepts of factors and multiples.

5. Problem 2
 36 is a multiple of 12
 What times 12 equals 36?
36= 12 x ___
3

6. Finding Factors- 8.1
 One way to find all the factors of a counting numbers is to
divide the number by all the counting numbers smaller than it
to see which numbers have equal answers. Whole number
quotients of the problem are usually factors .
 Find all the factors of M
1 x M = __
2 x M = __
3 X M = __
All Answers are
Factors of M

7. Problem 3
 : Find all the factors of 40
Start with the smallest numbers and work up
Divide 40 by 1, 2, 3, 4, 5 and so on until 40
1/40 = 40
2/40 = 20
3/40 = Doesn’t divide equally
4/40 = 10
5/40 = 8
 So 1,2,4,5,8,10,20,40 are all factors of 40.

8. Quick Notes- 8.1
 Every counting number except 1 must have at least two
distinct factors and itself.
 EX: 1 and 20 are factors of 20.

9. Finding Factors- 8.1
 The word factor has a double meaning, factor and to factor.
To factor a number A, can be described as A written as a
product of two or more counting numbers each of which is
less than A.
 EX: A as B can be factored C times because A= B x C

10. Problem 4
 How Many times can 18 be factored into 9?
 18 as 9 can be factored 2 times because 18 = 9 x 2.
 BUT 18 can also be factored further looking into the factors
of 9.
18 = 3 x 3 x 2.

11. Greatest Common Factor- 8.2
 When you have two or more counting numbers then the
Greatest Common Factor (GCF) is the greatest number or
factor that occurs in both numbers.
 The GCF is also known as the Greatest Common Divisor.

12. Problem 5
 What is the GCF of 12 and 18?
 First find all the factors of both numbers
12= 1,2,3,4,6,12
18= 1,2,3,6,9,18
 Then compare and find similar numbers
12= 1,2,3,4,6,12
18= 1,2,3,6,9,18
 The GCF of 12 and 18 is 6

13. Least Common Multiple- 8.2
 The Least Common Multiple (LCM) is the lowest number
that is a multiple of both numbers A and B.
 LCM of A and B
 List all the multiples of each number by multiplying by 1,2,3…
 Find all the common numbers
 The lowest multiple is the answer

14. Problem 6
 What is the LCM of 6 and 8?
 Determine the Multiples.
6= 6, 12, 18, 24, 30,36, 42, 48, 54, 60, 66, 72, …
8= 8,16, 24,32,40,48, 56, 64, 72, 80, …
 Find all the common multiples
24, 48, 72
 The LCM of 6 and 8 is 24.

15. Using GCF’s with FRACTIONS- 8.2
 To make a fraction into its simplest form, we divide the
numerator and the denominator of the fraction by the 2
numbers,

16. Problem 7
 Find in its simplest form through GCFs.
 Find the GCF of 24 and 36 = 12
 Now figure out what number multiplied by 12 equals 36 and 24
24= 2 x 12 36= 3 x 12
 Now take the two smallest numbers in each multiplication
equation and place them where their sums would be
24= 2 x 12 36= 3 x 12
 = Simplest form

17. Using LCM’s with FRACTIONS- 8.2
 The LCM of two fractions can be used when adding
the two fractions a common denominator
must be found.
 This makes the problem easier and prevents simplifying at
the end, a LCM is the easiest approach.

18. Problem 8

 What is the LCM of 6 and 8?
The number with the least value that is divisible by 6 and 8
= 24
 Now what numbers go into 6 and 8 to equal 24…4 and 3
 Next take the largest sums and add them together.
 is the sum