Factors And Multiples

In this article, the lesson objectives are: 

1. Factors
2. Multiples

What are Factors?

Factors are numbers that can divide another number completely without any remainder.

Example: 

Since 10 can be divided by 2, we say that 2 is a factor of 10.

In other words, factors of a number are divisors and quotients of the same number. Take note that the remainder must be 0.

5 and 2 are factors of 10. 

Is 3 a factor of 10? 

3 is not a factor of 10 because when we divide 10 by 3, we get 1 as the remainder.

\(10 ÷ 3 = 3 \text{ R } 1 \)

What are the other factors of 10? 

\(10 = 1 \times 10\)

     \( = 2 \times 5\)

Therefore, factors of 10 are 1, 2, 5 and 10.

Question 1: 

List all the factors of 24.

Solution: 

To find all the factors of 24, we can write down pairs of numbers that multiply to give us 24. 

We will begin with the smallest whole number which is 1. 

\(\begin{align}​​ 24 &= 1 \times 24\\ &= 2 \times 12\\ &= 3 \times 8\\ &= 4 \times 6  \end{align}\)

Factors of 24: 1, 2, 3, 4, 6, 8, 12 and 24

Answer:

1, 2, 3, 4, 6, 8, 12 and 24

Question 2: 

Which of the following is a factor of 84?

1. 5
2. 7
3. 8
4. 9

Solution: 

\(\begin{align}​​ 84 &= 1 \times 84\\ &= 2 \times 42\\ &= 3 \times 28\\ &= 4 \times 21\\ &= 6 \times 14\\ &= 7 \times 12 \end{align} \)

Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84

OR

To know which of the above is a factor of 84, we can divide 84 by each of them and check which number does not give us a remainder. 

\(\begin{align}​​ 84 \div 5 &= 16 \text{ R } 4\\ 84 \div 7 &= 12\\ 84 \div 8 &= 10 \text{ R } 4\\ 84 \div 9 &= 9 \text{ R } 3 \end{align}\)

Hence, only 7 can be divided by 84 with no remainder. 

Answer:

(2) 7

Question 3: 

Find the product of all the factors of 15.

1. 24
2. 45
3. 75
4. 225

Solution: 

\(\begin{align}​​ 15 &= 1 \times 15\\ &= 3 \times 5\ \end{align}\)

Factors of 15: 1, 3, 5 and 15

Product of all the factors of 15

\(\begin{align}​​ &= 1 \times 3 \times 5 \times 15 \\ &= 225 \end{align}\)

Answer:

(4) 225

Question 4: 

What are the common factors of 35 and 60?

Solution: 

Step 1: 

List down the factors of 35 and factors of 60. 

\(\begin{align}​​ 35 &= 1 \times 35\\ &= 5 \times 7 \end{align}\)

Factors of 35: 1, 5, 7 and 35

\(\begin{align}​​ 60 &= 1 \times 60\\ &= 2 \times 30\\ &= 3 \times 20\\ &= 4 \times 15\\ &= 5 \times 12\\ &= 6 \times 10 \end{align}\)

Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60

Step 2: 

Look at the factors of 35 and 60. If a number appears in the factors of 35 and in the factors of 60, it is a common factor of 35 and 60. 

Factors of 35: 1, 5, 7 and 35

Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60

Common factors of 35 and 60 are 1 and 5.

Answer:

1 and 5

Question 5: 

Find the greatest common factor of 42 and 56.

1. 1
2. 2
3. 7
4. 14

Solution: 

Step 1:

List the factors of 42 and the factors of 56.

\(\begin{align}​​ 42 &= 1 \times 42\\ &= 2 \times 21\\ &= 3 \times 14\\ &= 6 \times 7 \end{align}\)

Factors of 42: 1, 2, 3, 6, 7, 14, 21 and 42

\(\begin{align}​​ 56 &= 1 \times 56\\ &= 2 \times 28\\ &= 4 \times 14\\ &= 7 \times 8 \end{align}\)

Factors of 56: 1, 2, 4, 7, 8, 14, 28 and 56

Step 2:

Find the common factors of 42 and  56.

Factors of 42: 1, 2, 3, 6, 7, 14, 21 and 42

Factors of 56: 1, 2, 4, 7, 8, 14, 28 and 56

Common factors of 42 and 56: 1, 2, 7 and 14

Greatest common factor: 14

Answer:

(4) 14

What are Multiples?

A multiple is the product when one number is multiplied by another number. The numbers in the multiplication tables of the numbers are the multiples. 

Example: 

1st multiple of 6: \(1 \times 6 = 6 \)

2nd multiple of 6: \(2 \times 6 = 12 \)

3rd multiple of 6: \(3 \times 6 = 18 \)

and so on.

Question 1: 

List the first 3 multiples of 9.

Solution: 

Multiples of 9: 9, 18, 27 …

Answer:

9, 18 and 27

Question 2: 

65 is the __________ multiple of 5. 

1. 13th
2. 15th
3. 225th
4. 325th

Solution: 

To find which multiple of 5 is 65, we divide 65 by 5.

\(65 = 5 \;\times\) _________

\(65 ÷ 5 = 13\)

Answer:

(1) 13

Question 3: 

Find the smallest common multiple of 4, 6 and 9.

Solution: 

Step 1:

List the multiples of 4, multiples of 6 and multiples of 9. 

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44 

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48 

Multiples of 9: 9, 18, 27, 36, 45, 54 

Step 2: 

Look at the multiples of 4, 6 and 9. If a number appears in all 3 lists of multiples, it is a common multiple of 4, 6 and 9. 

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44 

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48 

Multiples of 9: 9, 18, 27, 36, 45, 54 

The smallest common multiple of 4, 6 and 9 is 36.

Answer:

36

Question 4: 

Find the product of the 3rd multiple and the 5th multiple of 6.

Solution: 

3rd multiple of 6

\(\begin{align}​​ &= 6 \times 3\\ &= 18 \end{align} \)

5th multiple of 6

\(\begin{align}​​ &= 6 \times 5\\ &= 30 \end{align} \)

Product of  3rd multiple and the 5th multiple of 6

\(\begin{align}​​ &= 18 \times 30\\ &= 540 \end{align} \)

OR

Product of  3rd multiple and the 5th multiple of 6

\(\begin{align}​​ &= 6 \times 3 \times 6 \times 5 \\ &= 18 \times 6 \times 5 \\ &= 108 \times 5\\ &= 540 \end{align}\)

Answer:

540

Question 5: 

The 3rd multiple of 6 has the same value as the 9th multiple of __________.

1. 18
2. 2
3. 27
4. 162

Solution:  

3rd multiple of 6

\(\begin{align}​​ &= 6 \times 3\\ &= 18\\[3ex] &18 \div 9 = 2 \end{align} \)

Answer:

(2) 2