Class 5 - Factors

Factors are essential in our everyday lives. We frequently use factors to arrange objects in rows and columns, to find whether objects can be split into smaller groups, etc.

Table of Contents

• What are the Factors?
• Factors of Prime and Composite Numbers
• Properties of Factors
• How to Find Factors of a Number?
• Finding Factors using Multiplication
• Finding Factors Using Division
• Factor Pairs
• Solved Problems on Factors
• Practice Questions on Factors

What are the Factors?

A factor is a number that divides another number without leaving a remainder. When you multiply two numbers together, the result is called a product. Each of those numbers is called a factor of that product. If you can share 12 oranges with a group of friends and have none left over, the number of people in that group is a factor of 12.

Factors show up in daily life more than you might think. When you arrange chairs in equal rows for a school event, or split a bill equally among friends, or reduce a fraction to its simplest form factors are at work behind the scenes.

Definition: A factor of a number N is any whole number that divides N exactly, leaving a remainder of zero.

Each arrangement above uses all 12 objects in perfectly equal groups. That is exactly what a factor does it divides a number without leaving anything behind.

Factors of Prime and Composite Numbers

Numbers fall into two main groups based on how many factors they have.

Prime Number:

A prime number has exactly two factors the number 1 and the number itself. Nothing else divides it evenly. For example, 7 is a prime number. If you try dividing 7 by 2, 3, 4, 5, or 6, you always get a remainder. Only 1 and 7 divide it cleanly. Some common prime numbers are 2, 3, 5, 7, 11, 13, and 17.

Composite Number:

A composite number, on the other hand, has more than two factors. It can be divided evenly by at least one number other than 1 and itself. For example, 18 is a composite number. It can be divided evenly by 1, 2, 3, 6, 9, and 18 that is six factors in total.

One small exception worth knowing: the number 1 is neither prime nor composite. It has only one factor, which is 1 itself, so it sits in a category of its own.

Prime Number	Factors	Composite Number	Factors
2	1, 2	4	1, 2, 4
3	1, 3	6	1, 2, 3, 6
5	1, 5	9	1, 3, 9
7	1, 7	12	1, 2, 3, 4, 6, 12
11	1, 11	15	1, 3, 5, 15
13	1, 13	20	1, 2, 4, 5, 10, 20
17	1, 17	36	1, 2, 3, 4, 6, 9, 12, 18, 36

Note: 1 is neither prime or composite. It has only one factor itself.

Number	Type	Factors	Total Count
2	Prime	1, 2	2
7	Prime	1, 7	2
13	Prime	1, 13	2
4	Composite	1, 2, 4	3
12	Composite	1, 2, 3, 4, 6, 12	6
30	Composite	1, 2, 3, 5, 6, 10, 15, 30	8

Properties of Factors

• There are a few useful rules about factors that make working with them much easier:
• Every number has a limited, countable set of factors. You will never find infinitely many factors for any number the list always ends somewhere.
• The factor of a number is always smaller than or equal to the number itself. A factor cannot be bigger than the number it divides. For instance, no factor of 10 can be greater than 10.
• The number 1 is a factor of every single number that exists. And every number is a factor of itself.
• Apart from 0 and 1, every number has at least two factors 1 and the number itself. Numbers like 2, 3, 5, and 7 have exactly these two factors, which is why they are prime.
• Division and multiplication are the two operations used to find factors.

How to Find Factors of a Number?

There are two straightforward methods to find the factors of any number: the division method and the multiplication method. Both work well it just depends on which one feels more natural to you.

Finding Factors Using Division

In this method, you divide the given number by smaller numbers one at a time, starting from 1. Each time the division gives no remainder, you have found a factor.

Example: Find the factors of 18.

Start dividing 18 by 1, then 2, then 3, and keep going:

18 ÷ 1 = 18 (no remainder) 

1 is a factor 18 ÷ 2 = 9 (no remainder) 

2 is a factor 18 ÷ 3 = 6 (no remainder) 

3 is a factor 18 ÷ 4 = 4.5 (has decimal) 

4 is not a factor 18 ÷ 5 = 3.6 (has decimal) 

5 is not a factor 18 ÷ 6 = 3 (no remainder) 

6 is a factor 18 ÷ 9 = 2 (no remainder) 

9 is a factor 18 ÷ 18 = 1 (no remainder) 

18 is a factor

So the factors of 18 are: 1, 2, 3, 6, 9, and 18.

Finding Factors Using Multiplication

Here, instead of dividing, you look for pairs of numbers that multiply together to give the original number.

Example: Find the factors of 24.

Ask yourself: which pairs of whole numbers multiply to make 24?

1 × 24 = 24 → (1, 24)

2 × 12 = 24 → (2, 12)

3 × 8 = 24 → (3, 8)

4 × 6 = 24 → (4, 6)

From these pairs, collect all the individual numbers:

Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24.

Both methods lead you to the same answer. Choose whichever feels easier.

Factor Pairs

A factor pair is simply a set of two numbers that, when multiplied together, give the original number. Every number has at least one factor pair.

Factor pairs cannot be fractions or decimals they must be whole numbers.

Example: Find the factor pairs of 36.

First, list all factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.

Now pair them up so each pair multiplies to give 36:

1 × 36 = 36 → pair is (1, 36)

2 × 18 = 36 → pair is (2, 18)

3 × 12 = 36 → pair is (3, 12)

4 × 9 = 36 → pair is (4, 9)

6 × 6 = 36 → pair is (6, 6)

So the factor pairs of 36 are: (1, 36), (2, 18), (3, 12), (4, 9), and (6, 6).

Solved Problems on Factors

Problem 1: Find all the factors of 30.

Using the division method: 30 ÷ 1 = 30 

30 ÷ 2 = 15 

30 ÷ 3 = 10 

30 ÷ 5 = 6 

30 ÷ 6 = 5 

30 ÷ 10 = 3 

30 ÷ 15 = 2 

30 ÷ 30 = 1 

Factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30.

Problem 2: Is 9 a factor of 54?

Divide 54 by 9: 54 ÷ 9 = 6.

There is no remainder.

Yes, 9 is a factor of 54.

Problem 3: How many factors does 20 have?

Factors of 20: 1, 2, 4, 5, 10, 20.

20 has 6 factors in total.

Problem 4: Find the factor pairs of 48.

1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8.

Factor pairs of 48 = (1, 48), (2, 24), (3, 16), (4, 12), (6, 8).

Practice Questions on Factors

1. Write all the factors of 45.
2. Find the factors of 60 using the multiplication method.
3. List the factor pairs of 56.
4. Is 7 a factor of 49? Give a reason.
5. Which number among 4, 5, 6, and 7 is a factor of 42?
6. How many factors does the number 36 have?
7. Find all the common factors of 24 and 36.
8. A teacher wants to arrange 40 students into equal rows. What are the possible row sizes? (Hint: find factors of 40.)