Class Notes and Practice Problems

Chapter 8Factors & Multiplication

Michelle LiSanti and Kate Muller

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.

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.

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 of A, B, and C of counting numbers if A = 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.

Problem 2

36 is a multiple of 12

What times 12 equals 36?

36= 12 x ___

3

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

Problem 3

: Find all the factors of 40Start with the smallest numbers and work upDivide 40 by 1, 2, 3, 4, 5 and so on until 40

1/40 = 402/40 = 203/40 = Doesn’t divide equally4/40 = 105/40 = 8

So 1,2,4,5,8,10,20,40 are all factors of 40.

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.

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

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.

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.

Problem 5

What is the GCF of 12 and 18? First find all the factors of both numbers

12= 1,2,3,4,6,1218= 1,2,3,6,9,18

Then compare and find similar numbers12= 1,2,3,4,6,1218= 1,2,3,6,9,18

The GCF of 12 and 18 is 6

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

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 multiples24, 48, 72

The LCM of 6 and 8 is 24.

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,

€

A

B

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

€

13

24=2

3

€

24

36

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.€

A

B+C

D

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

€

1

6+3

8

€

1

6x4

4=4

24→3

8x3

3=9

24

€

4

24+9

24= ?

€

13

24