# Factors & Multiples

Factors and Multiples We learn how to find the multiples of whole numbers as well as how to find a number's factors.

## Definition - Multiples

Given an integer, for example $$3$$, the multiples of $$3$$ are all the integers we obtain if we multiply $$3$$ by the positive integers: $$1$$, $$2$$, $$3$$, ... .

So the multiples of $$3$$ are: \begin{aligned} & 1 \times 3 = 3\\ & 2 \times 3 = 6 \\ & 3 \times 3 = 9 \\ & 4 \times 3 = 12\\ & \dots \end{aligned} A more general definition of the multiples of an arbitrary integer $$n$$ would be: $k \times n$ Where $$k$$ is a positive integer.

## Tutorial 1: Multiples

In the following tutorial we review the ... .

## Exercise 1

1. Write down the first $$5$$ multiples of $$3$$.

2. What is the $$7^{\text{th}}$$ multiple of $$2$$?

3. State the first $$4$$ multiples of $$7$$.

4. What is the $$6^{\text{th}}$$ multiple of $$4$$?

5. Write down the first $$6$$ multiples of $$5$$.

1. The first $$5$$ multiples of $$3$$ are: $$3, 6, 9, 12, 15$$

2. The seventh multiple of $$2$$ is $$14$$.

3. The first $$4$$ multiples of $$7$$ are: $$7,14,21,28$$.

4. The $$6^{\text{th}}$$ multiple of $$4$$ is $$24$$.

5. The first $$6$$ multiples of $$5$$ are: $$5,10,15,20,25,30$$.

## Solution With Working

1. The first $$5$$ multiples of $$3$$ are:

• $$1\times 3 = 3$$

• $$2\times 3 = 6$$

• $$3\times 3 = 9$$

• $$4\times 3 = 12$$

• $$5\times 3 = 15$$

2. The $$7^{\text{th}}$$ multiple of $$2$$ is: $$7\times 2 = 14$$

3. The first $$4$$ multiples of $$7$$ are:

• $$1\times 7 = 7$$

• $$2\times 7 = 14$$

• $$3\times 7 = 21$$

• $$4\times 7 = 28$$

4. The $$6^{\text{th}}$$ multiple of $$4$$ is: $$6 \times 4 = 24$$.

5. The first $$6$$ multiples of $$5$$ are:
• $$1\times 5 = 5$$

• $$2\times 5 = 10$$

• $$3\times 5 = 15$$

• $$4\times 5 = 20$$

• $$5\times 5 = 25$$

• $$6\times 5 = 30$$

## Exercise 2

State whether each of the following statements is true or false.

1. Statement: $$12$$ is a multiple of $$3$$.

2. Statement: $$7$$ is a multiple of $$1$$.

3. Statement: The $$4^{\text{th}}$$ multiple of $$5$$ is $$20$$.

4. Statement: The $$3^{\text{rd}}$$ multiple of $$7$$ is $$23$$.

5. Statement: The $$3^{\text{rd}}$$ multiple of $$4$$ equals to the $$4^{\text{th}}$$ multiple of $$3$$.

1. true

2. true

3. true

4. false

5. true

## Solution With Working

1. True, since $$12 = 4\times 3$$. $$12$$ is the $$4^{\text{th}}$$ multiple of $$3$$.

2. True, since $$7 = 7 \times 1$$. $$7$$ is the $$7^{\text{th}}$$ multiple of $$1$$.

3. True, since $$4\times 5 = 20$$, the $$4^{\text{th}}$$ multiple of $$5$$ is $$20$$.

4. False, the $$3^{\text{rd}}$$ multiple of $$7$$ is $$3\times 7 = 21$$, which isn't equal to $$23$$.

5. True, the $$3^{\text{rd}}$$ multiple of $$4$$ is $$3\times 4 = 12$$ and the $$4^{\text{th}}$$ multiple of $$3$$ is $$4\times 3 = 12$$.

## Definition - Factors

Given an integer $$p$$, we say $$n$$ is a factor of $$p$$ if and only if $$p$$ is a multiple of $$n$$.

For instance, $$3$$ is a factor of $$6$$ because $$6$$ is a multiple of $$3$$.

Indeed we can write: $6 = 2\times 3$ Note: That also shows us that $$2$$ is a factor of $$6$$ because $$6$$ is a multiple of $$2$$, which highlights an important fact: we find factors in pairs. In other words, we always find factors two at a time.

## Tutorial 2: Factors

In the following tutorial we learn what a factor is as well as see how to find a number's factors.

## Exercise 3

Using the method we just learnt, find the following factors:

1. factors of $$8$$

2. factors of $$9$$

3. factors of $$10$$

4. factors of $$12$$

5. factors of $$14$$

6. factors of $$20$$

## Solution Without Working

1. The factors of $$8$$ are: $1,2,4,8$

2. The factors of $$9$$ are: $1,3,9$

3. The factors of $$10$$ are: $1,2,5,10$

4. The factors of $$12$$ are: $1,2,3,4,6,12$

5. The factors of $$14$$ are: $1,2,7,14$

6. The factors of $$20$$ are: $1,2,4,5,10,20$

## Solution With Working

1. To find the factors of $$8$$ we work our way through the whole numbers less than or equal to $$8\div 2 = 4$$ and check if $$8$$ is a multiple:
• $$1$$: $$8$$ is a multiple of $$1$$, indeed: $$8 = 8\times 1$$, so $$1$$ is a factor of $$8$$ and so is $$8$$.

• $$2$$: $$8$$ is a multiple of $$2$$, indeed: $$8 = 4\times 2$$, so $$2$$ is a factor of $$8$$ and so is $$4$$.

• $$3$$: $$8$$ is not a multiple of $$3$$, so $$3$$ is not a factor of $$8$$.

• $$4$$: just as we saw, when considering the factor $$2$$, $$4$$ is a factor of $$8$$.

So the factors of $$8$$ are: $$1,2,4,8$$.

2. To find the factors of $$9$$, we work our way through the whole numbers less than or equal to $$9 \div 2 = 4.5$$, those are the whole numbers from $$1$$ to $$4$$, and check if $$9$$ is a multiple:
• $$1$$: $$9$$ is a multiple of $$1$$, indeed: $$9 = 9 \times 1$$, so $$1$$ is a factor of $$9$$ and so is $$9$$.

• $$2$$: $$9$$ isn't a multiple of $$2$$, so $$2$$ isn't a factor of $$9$$.

• $$3$$: $$9$$ is a multiple of $$3$$, indeed: $$9 = 3\times 3$$, so $$3$$ is a factor of $$9$$.

• $$4$$: $$9$$ isn't a mutiple of $$4$$ so $$4$$ isn't a factor of $$9$$.

So the factors of $$9$$ are $$1$$, $$3$$ and $$9$$.

3. To find the factors of $$10$$, we work our way through the whole numbers less than or equal to $$10 \div 2 = 5$$, those are the whole numbers from $$1$$ to $$5$$, and check if $$10$$ is a multiple:
• $$1$$: $$10$$ is a multiple of $$1$$, indeed $$10 = 10\times 1$$, so $$1$$ is a factor and so is $$10$$.

• $$2$$: $$10$$ is a multiple of $$2\à, indeed \(10 = 5 \times 2$$, so $$2$$ is a factor of $$10$$ and so is $$5$$.

• $$3$$: $$10$$ isn't a multiple of $$3$$ so $$3$$ isn't a factor of $$10$$.

• $$4$$: $$10$$ isn't a multiple of $$4$$ so $$4$$ isn't a factor of $$10$$.

• $$5$$: jsut as we saw, when studying the factor $$2$$, $$5$$ is a factor of $$10$$.

So the factors of $$10$$ are $$1,2,5$$ and $$10$$.

4. To find the factors of $$12$$, we work our way through the whole numbers less than or equal to $$12 \div 2 = 6$$, those are the whole numbers from $$1$$ to $$6$$, and check if $$12$$ is a multiple:
• $$1$$: $$12$$ is a multiple of $$1$$, indeed $$12 = 12 \times 1$$, so $$1$$ is a factor of $$12$$ and so is $$12$$.

• $$2$$: $$12$$ is a multiple of $$2$$, indeed $$12 = 6 \times 2$$, so $$2$$ is a factor of $$12$$ and so is $$6$$.

• $$3$$: $$12$$ is a multiple of $$3$$, indeed $$12 = 4 \times 3$$, so $$3$$ is a factor of $$12$$ and so is $$4$$.

• $$4$$: just as we saw, when studying the factor $$3$$, $$4$$ is a factor of $$12$$

• $$5$$: $$12$$ is not a multiple of $$5$$, so $$5$$ is not a factor or $$12$$.

• $$6$$: just as we saw, when studying the factor $$2$$, $$6$$ is a factor of $$12$$

So the factors of $$12$$ are $$1,2,3,4,6$$ and $$12$$

5. To find the factors of $$14$$, we work our way through the whole numbers less than or equal to $$14 \div 2 = 7$$, those are the whole numbers from $$1$$ to $$7$$, and check if $$14$$ is a multiple:
• $$1$$: $$14$$ is a multiple of $$1$$, indeed $$14 = 14\times 1$$, so $$1$$ is a factor of $$14$$ and so is $$14$$.

• $$2$$: $$14$$ is a multiple of $$2$$, indeed $$14 = 7 \times 2$$, so $$2$$ is a factor of $$14$$ and so is $$7$$

• $$3$$: $$14$$ is not a multiple of $$3$$ so $$3$$ is not a factor of $$14$$.

• $$4$$: $$14$$ is not a multiple of $$4$$ so $$4$$ is not a factor of $$14$$.

• $$5$$: $$14$$ is not a multiple of $$5$$ so $$5$$ is not a factor of $$14$$.

• $$6$$: $$14$$ is not a multiple of $$6$$ so $$6$$ is not a factor of $$14$$.

• $$7$$: just as we saw, when studying the factor $$2$$, $$7$$ is a factor of $$14$$.

So the factors of $$14$$ are $$1,2,7$$ and $$14$$.

6. To find the factors of $$20$$, we work our way through the whole numbers less than or equal to $$20 \div 2 = 10$$, those are the whole numbers from $$1$$ to $$10$$, and check if $$20$$ is a multiple:
• $$1$$: $$20$$ is a multiple of $$1$$, indeed $$20 = 20\times 1$$, so $$1$$ is a factor of $$20$$ and so is $$20$$.

• $$2$$: $$20$$ is a multiple of $$2$$, indeed $$20 = 10\times 2$$, so $$2$$ is a factor of $$20$$ and so is $$10$$.

• $$3$$: $$20$$ is not a multiple of $$3$$ so $$3$$ is not a factor of $$20$$.

• $$4$$: $$20$$ is a multiple of $$4$$, indeed $$20 = 5 \times 4$$, so $$4$$ is a factor of $$20$$ and so is $$5$$.

• $$5$$: as we have just seen, when studying the factor $$4$$, $$5$$ is a factor of $$20$$.

• $$6$$: $$20$$ is not a multiple of $$6$$ so $$6$$ is not a factor of $$20$$.

• $$7$$: $$20$$ is not a multiple of $$7$$ so $$7$$ is not a factor of $$20$$.

• $$8$$: $$20$$ is not a multiple of $$8$$ so $$8$$ is not a factor of $$20$$.

• $$9$$: $$20$$ is not a multiple of $$9$$ so $$9$$ is not a factor of $$20$$.

• $$10$$: as we saw, when considering the factor $$2$$, $$10$$ is a factor of $$20$$.

So the factors of $$20$$ are $$1,2,4,5,10$$ and $$20$$.

## Exercise 4

State whether each of the following statements is "true" or "false":

1. $$9$$ is a factor or $$12$$.

2. $$12$$ is a factor of $$3$$.

3. $$3$$ is a factor of $$12$$.

4. $$8$$ is a factor of $$24$$.

5. $$1$$ is a factor of all whole numers.

6. If $$n$$ is a factor of $$m$$ then $$m$$ is a multiple of $$n$$.

1. false

2. false

3. true

4. true

5. true

6. true

## Solution With Working

1. false: $$9$$ is not a factor of $$12$$ since $$12$$ is not a multiple of $$9$$.

2. false: $$12$$ is not a factor of $$3$$, it is a multiple of $$3$$.

3. true: $$3$$ is a factor of $$12$$ since $$12$$ is a multiple of $$3$$.

4. true: $$8$$ is a factor of $$24$$ since $$24$$ is a multiple of $$8$$.

5. true: $$1$$ is a factor of all whole numbers since all whole numbers are multiples of $$1$$.

6. true: by definition a whole number $$n$$ is a factor of another whole number $$m$$ if and only if $$m$$ is a multiple of $$n$$. So if $$n$$ is a factor of $$m$$ then $$m$$ must be a multiple of $$n$$.